Off-axis reflective optical systems for freeform surfaces have the advantages of no chromatic aberration, no central obscuration, and a large field of view in one direction, so they are widely used in many optical observation fields. The optical system design and machining, as well as precise system assembly, are three key core technologies to obtain high quality optical imaging systems. Current design methods for freeform optical systems rely on the experience of optical designers to obtain initial structural parameters before calculating the coordinates of the discrete points on a mirror surface. To solve this problem, it is necessary to design a fast and accurate method for obtaining the initial structural parameters of an off-axis reflective optical system during the design. Furthermore, the optical alignment to assembly the manufactured system is also a high-cost and complex operation. Current integrated processing and manufacturing methods avoid the repeated assembly process of off-axis reflective systems using incorporating manufacturing constraints during the design. However, the manufacturing constraints are not considered by most design methods for freeform off-axis reflective systems, resulting in the designed system being unable to meet the requirements of integrated machining and manufacturing. Therefore, design methods are needed to guide the design of initial structural parameters and freeform off-axis reflective optical system under manufacturing constraints.

We propose an automatic generation method to design the starting points for off-axis freeform systems based on manufacturing constraints. First, the unified description of the surface shape and pose of mirrors in a global coordinate system is provided and the method for ray tracing is given. Second, the manufacturing constraint module is designed. When designing the initial structure of the off-axis system based on manufacturing constraints, it is necessary to take into account both the removal of the obstruction and the fit degree of the mirrors to the cylindrical reference surface. Therefore, a manufacturing constraint module composed of a degree of co-circularity function and obstruction evaluation function is proposed to assess the rationality of the initial structure. Then, we conduct a comprehensive objective function for the initial structure of the optical system by tracing the optical path structure of the off-axis reflective system. The most suitable initial structure parameters for the design requirements are achieved by searching the minimum value of the comprehensive objective function. Finally, taking the initial structural parameters as the input, combined with the improved Wassermann-Wolf (W-W) method, we propose a design method for a freeform off-axis reflective system under manufacturing constraints.

Two freeform off-axis three-mirror reflective systems are designed by the proposed method. The field of view of the first system is 4°×4° and the F-number is 3.3. The entrance pupil diameter is 90 mm and the radius of the cylindrical reference surface is 150 mm. The initial structural parameters are searched based on these design requirements, and the starting point of the freeform off-axis three-mirror system is obtained through the combination of the improved W-W method (Fig. 5). To further quantify and verify the performance of the designed optical system, we change the tilt angle and mirror spacing of the mirrors. In addition, 10 different mirror pose combinations are randomly generated to simulate the possible combinations of mirror positions, which are used by the improved W-W method directly without searching for the initial structural parameters. The imaging quality of these 10 systems has significant uncertainty, and the average manufacturing error value is higher than the design starting point generated by the proposed method. The field of view of the second system is 2°×2°, the F-number is 3.5, and the entrance pupil diameter is 85 mm. In the generated design starting point, the three mirrors are close to the cylindrical reference plane, and the rays of each field of view converge at the image plane. After obtaining 10 random mirror position combinations, the obstruction is manually removed to simulate the possible mirror position combinations used to generate the design starting points directly by the improved W-W method when the initial structural parameters were obtained by manual off-axis. Among the 10 design starting points generated directly by the improved W-W method, the average root mean square (RMS) wavefront error is 0.95λ (λ is wavelenth) and the average RMS spot radius is 36.10 μm (Fig. 10), both are higher than the design starting point generated by the proposed method, and the imaging qualities are unstable. This indicates that even in the absence of obstruction in the initial structure, the RMS wavefront error and RMS spot radius of the system generated directly by the improved W-W method are strongly affected by the changes in the distance between mirrors, and the manufacturing error is still greater than the design starting point generated using the proposed method.

We propose a design method for off-axis reflective optical systems with freeform surfaces under manufacturing constraints. Under the consideration of manufacturing constraints, the initial structural parameters that meet the design requirements are automatically obtained through simple searching, and then the design starting point of the freeform off-axis reflective system is obtained. The experimental results show that by searching the minimum values of the comprehensive objective function proposed in this study and passing these values into the improved W-W method, the co-circularity characteristics can be significantly improved with no optical path blocking, and the problem of unstable imaging quality is avoided. The proposed method can effectively guide the freeform off-axis reflective systems manufactured by the integrated ultra-precision grating milling technology, and also can be applied to design more general reflective systems. The designers can choose or change the range of application of the subfunctions in the objective function.