# A Design for a Manufacturing-Constrained Off-Axis Four-Mirror Reflective System
## 2 Working Principle  
### Manufacturing Constraints Analysis for Four-Mirror System  
Manufacturing constraints in optical systems depend on practical manufacturing methods. Figure 1a demonstrates a computer integrated manufacturing system for an off-axis four-mirror reflective optical system based on raster milling. In order to avoid the time-consuming machining process and the tedious assembly process before the system is put into use, all the mirror blanks can be assembled according to the design position, and then all the mirrors can be processed synchronously by raster milling. In this method of processing, all mirrors are free of disassembly and assembly, in which the position accuracy of the mirrors depends only on the machine tool. Compared with the traditional method, there is no need to process the locating surface so that the alignment error can be avoided, and the system can be put into use after the processing is completed. An ultra-precision machine tool has a high motion accuracy, which ensures the excellent position accuracy between the mirrors. Therefore, the good optical performance of the whole system is ensured. The curvature radii of the mirror surfaces in the system should be greater than the rotational radius of the tools, and all the mirror surfaces should be close to a cylindrical reference surface, as shown in Figure 1b. These manufacturing constraints should be reflected in the design process of the system to ensure a smooth machining process. In practice, there is always a deviation between the mirror and the reference surface. For one mirror in this system, the region between the mirror's surface and the reference surface is called the deviation region, as shown in Figure 2.  
To quantify this deviation, the index called the average deviation from the reference surface (ADRS), \(\sigma_{\text{ADRS}}\), is defined to characterize the average deviation between the mirrors' surface and the reference surface in the design,  
\[\sigma_{\text{ADRS}}=\frac{1}{\Omega}\int\limits_{\Omega}|S-R|d\Omega, \tag{1}\]  
Figure 1: (**a**) Machining configuration for the four-mirror system. (**b**) The geometric relationship between the mirrors’ surfaces and the reference surface. In the figure, \(O\) is the center of the reference surface and \(R\) is the radius of the reference surface. Figure 2: The deviation region between the mirror’s surface and the reference surface. In the figure, \(\Omega\) is the integral region and \(\theta\) is the angle formed by the mirror’s edge and the Z-axis of the cylindrical coordinate system. where \(S\) is the equation of the mirror's surface in the cylindrical coordinate system and \(R\) is the radius of the reference surface. \(\Omega\) is the integral region, which has the following differential form:  
\[d\Omega=d\theta dz, \tag{2}\]  
where \(\theta\) is the angle formed by the mirror's edge and the Z-axis of the cylindrical coordinate system. In the raster milling process, the deviation region of a cross-section perpendicular to the tool's rotation direction should also be considered, as shown in Figure 3a. In a cross-section of this structure, Equation (1) can be rewritten as  
\[\sigma_{\text{ADBS}}=\frac{1}{\theta}\int|\rho-R|d\theta, \tag{3}\]  
where \(\theta\) is the central angle of the mirror, \(\rho\) is the equation of the mirror's surface in the polar coordinate system and \(R\) is the radius of the reference surface. Equation (3) also applies to other possible positional relationships of the mirror and the reference surface, as shown in Figure 3b. This index can effectively reflect the deviation degree regardless of the different positions and sizes of mirrors. Meanwhile, the maximum deviation from the reference surface of each mirror also needs to be considered. In order to make this machining method easier to achieve, it is necessary to plan the tool's radius of gyration and tool spindle path in the design stage.