# Design of Parabolic Off-Axis Reflector Optical System for Large Aperture Single Star Simulators
## 3 Initial Structural Parameter Calculation and Design Optimization Process  
### Calculation of System Structure Parameters  
When designing the off-axis reflective system, it is necessary to design the co-axial system with shielding because the center is not shielded, as shown in the formula in Figure 2. According to the design index requirements, the optical system's initial structural parameters are solved using the primary aberration theory. For the center of the obstruction, the field of view or aperture diaphragm is off-axis to keep it away from the obstruction. Finally, the off-axis reflective system without obstruction is obtained by optimizing the system [27]. The primary mirror of this system uses a paraboloid, which is a quadratic surface, and the expression of the quadratic surface is shown in Equation (3). \[y^{2}=2R_{0}x-\Big{(}1-e^{2}\Big{)}x \tag{3}\]  
In Equation (3), \(R_{0}\) is the radius of curvature of the mirror vertex; \(e^{2}\) is the face shape parameter of the reflector, which can be used to eliminate aberration when used as an optimization variable. The collimation system of a single star simulator is meant to simulate stars at infinity. Therefore, the design structure of the telescope system can be used in the design of the star point simulation optical system. Thus, the single star simulator optical system satisfies the following two conditions: \begin{table}
\begin{tabular}{c c} \hline \hline
**System Parameter** & **Performance Index** \\ \hline Aperture & \(\Phi\)500 mm \\ Focus length & 5000 mm \\ Field of view & 0.01\({}^{\circ}\) \\ Spectral range & 450 nm-850 nm \\ Central wavelength & 632.8 nm \\ Wave aberration & \(\leq 1/25\lambda\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optical system design requirements. Figure 2: Initial structure of the co-axial system. (a) The object is located at infinity, \(l_{1}=\infty\), \(u_{1}=0\);  
(b) The diaphragm is located on the primary mirror, \(x_{1}=y_{1}=0\). According to Figure 2, the expressions for the blocking ratio \(\alpha\) of the system and the magnification \(\beta\) of the secondary mirror are shown in Equation (4). \[\left\{ \begin{array}{l}\alpha=\frac{l_{2}}{l_{1}}\approx\frac{h_{2}}{h_{1}} \\ \beta=\frac{l_{2}^{\prime}}{l_{2}}\approx\frac{f^{\prime}}{f_{1}^{\prime}} \end{array} \right. \tag{4}\]  
where \(h_{1}\) and \(h_{2}\) denote half of the aperture of the primary and secondary mirrors, respectively; \(-f_{1}{}^{\prime}\) and \(f_{1}{}^{\prime}\) denote the focal length of the primary mirror and the focal length of the star simulation optical system, respectively; \(l_{2}\) and \(l_{2}{}^{\prime}\) denote the distance between the secondary mirror and the object and the distance between the secondary mirror and the image, respectively. According to the structural design of the co-axial system, the expression of the secondary mirror object distance \(l_{2}\) can be known from the structure of the two-mirror system, as shown in Equation (5). \[l_{2}=\frac{(-f_{1}{}^{\prime}+\Delta)}{(\beta-1)} \tag{5}\]  
where \(\Delta\) is the focal point protrusion of the optical system. Since the shade ratio of the system will be directly affected by the focal point outreach of the system, after comprehensive consideration, the focal point outreach can be temporarily determined as \(\Delta=200\) mm. Knowing the relationship between the radius of curvature \(R_{1}\) of the primary mirror's vertex and the primary mirror's focal length, the radius of curvature \(R_{2}\) of the vertex of the secondary mirror is calculated using [PERSON]'s formula, as shown in Equation (6). \[\left\{ \begin{array}{l}R_{1}=2f_{1}{}^{\prime}\\ R_{2}=\frac{\alpha\beta}{1+\beta}R_{1}\end{array} \right. \tag{6}\]  
According to the primary phase aberration theory, the expressions for spherical aberration (\(S_{1}\)) and wise aberration (\(S_{2}\)) in a reflective optical system are shown in Equation (7). \[\left\{