# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 1 Introduction  
Reflective optical systems have attracted much interest because of their extensive use in many fields such as space optical systems [1] and infrared scanning systems [2, 3]. Due to the advantages of the reflective optical system such as being free of chromatic aberrations, a foldable optical path, a large aperture, insensitive to temperature and pressure changes, etc., more attention has been paid to the design of the reflective systems than to the refractive systems. In addition, making the field of view (FOV) off-axis can also solve the obstruction problem in reflective systems [4]. Among them, the four-mirror system can obtain a good imaging quality, a compact structure, a large FOV, and a large aperture because of its higher design degrees of freedom. With the development of the ultra-precision machining system, it is possible to manufacture reflective optical systems with complex surfaces such as aspheric surfaces and freeform surfaces [5, 6, 7]. The reflective optical systems usually go through a series of processes (e.g., design, machining, assembling, etc.) before being put into use. However, most of the current design methods focus only on how to improve the optical performance of these systems, while the manufacturing issues are not well considered in the design process. The consequence is that the optical system with complex surfaces is usually difficulty to achieve in machining and assembling, even if it may satisfy the optical requirements in theory. This reduces the practicability of the designed optical system. Introducing manufacturing constraints into the design process helps to simplify the machining and assembling process. [PERSON] et al. developed an off-axis three-mirror system with the primary mirror and tertiary mirror integrated on a single substrate, in which the number of mirrors that need optical alignment is reduced from three to two [8]. [PERSON] et al. proposed an alignment-free manufacturing approach tomachine the off-axis multi-reflective system, in which the measurement methods were also established to evaluate optical performance of the integrated system [9]. [PERSON] et al. proposed the concept of the reference surface to limit the position of each mirror in an off-axis three-mirror anastigmat (TMA) optical system [10]. The reference surface theory requires that the mirror's surface must be close enough to the reference surface to make the machining process more accessible. The shape of the reference surface depends on the machining method: for the turning and milling process, the reference surface are the symmetric rotational surface and the extended surface, respectively [11; 12; 13]. A cylindrical reference surface also helps to make the machining process more effective. When all mirror surfaces in a reflective optical system are close to the same cylindrical surface, the milling process will be greatly simplified, and the optical alignment caused by the asymmetry of the freeform surface can also be avoided. Therefore, the similarity between the mirror's surfaces and the reference surface should be considered at the level of design. At present, the mainstream optical design software adopts the damped least square (DLS) method as the optimization algorithm [14; 15]. This algorithm is a local optimization algorithm, which is easy to fall into the local minimum value in the optimization process, so the final design result is often the local optimal solution close to the initial point rather than the global optimal solution. Therefore, the existing optical design software depends heavily on the selection of the initial structure. In other words, the selection of a reasonable initial structure is a critical step in the optical system design. Generally, in order to design an optical system that meets the requirements, it is necessary to find the initial structure parameters from the existing references and then optimize the initial structure. However, for some optical systems with special structural configurations, this method may not be applicable.