## 2 Integrated Design Principles and Schemes
### Initial Structure of the Integrated System  
In order to make the visible light band and infrared band light path separated at the central opening of the primary mirror, and so that the light does not interfere with the components of the camera, the initial structure of the camera should be a coaxial three-mirror system with an intermediate image plane. The initial structure has seven free variables, where \(e_{1}{}^{2}\), \(e_{2}{}^{2}\), and \(e_{3}{}^{2}\) represent the quadric characteristics of the three mirrors, \(\beta_{1}\) is the magnification of the secondary mirror, \(\beta_{2}\) is the magnification of the third mirror, and \(\alpha_{1}\) is the obstruction ratio of the secondary mirror to the primary mirror, \(\alpha_{2}\) is the obstruction ratio of the third mirror to the secondary mirror. Since there is an intermediate image plane between the primary and secondary mirrors, \(\alpha_{2}<0\), \(\beta_{2}<0\); the beam emitted by the secondary mirror is a convergent beam, so \(\beta_{1}<0\); \(\alpha_{1}\) represents the distance between the primary and secondary mirrors, generally \(0<\alpha_{1}<1\). In addition, it is also necessary to control the distance \(\Delta\) between the intermediate image plane and the center of the primary mirror to better control the interference problem between the two subsystems. \(\Delta\) can be calculated by Equation (1), where \(l_{2}\) is the distance from the secondary mirror to the focal  
Figure 1: Schematic diagram of dual-band integrated system design. point of the primary mirror, and \(f_{1}{}^{\prime}\) is the focal length of the primary mirror [13]. The initial structure diagram of the three-mirror system is shown in Figure 2.  
\[\Delta=l_{2}(\beta_{1}-1)+f_{1}{}^{\prime} \tag{1}\]  
Through \(\alpha_{1}\), \(\alpha_{2}\), \(\beta_{1}\), and \(\beta_{2}\), the apex radius of curvature \(R_{1}\), \(R_{2}\), \(R_{3}\) of the primary mirror, secondary mirror and the third mirror, the distance \(d_{1}\) from the primary mirror to the secondary mirror, the distance \(d_{2}\) from the secondary mirror to the third mirror, and the distance \(l_{3}\)' between the third mirror to the image plane can be determined. The calculation equation refers to Equations (2)-(4). \[R_{1}=\frac{2f}{\beta_{1}\beta_{2}},\,R_{2}=\frac{2\alpha_{1}f}{\beta_{2}(1+ \beta_{1})},\,R_{3}=\frac{2\alpha_{1}\alpha_{2}f}{(1+\beta_{2})}. \tag{2}\]  
\[d_{1}=\frac{(1-\alpha_{1})f}{\beta_{1}\beta_{2}},\,d_{2}=\frac{(1-\alpha_{2}) \alpha_{1}f}{\beta_{2}}. \tag{3}\]  
\[l_{3}{}^{\prime}=f\alpha_{1}\alpha_{2}. \tag{4}\]  
According to the aberration elimination condition, Equations (5)-(8), the quadric coefficients of the three mirrors can be calculated. The three-mirror optical system can simultaneously eliminate four kinds of aberrations: spherical aberration, coma, astigmatism, and field curvature. \[ \begin{array}{c}S_{I}=0\\ \\ e_{2}^{2}(\alpha_{1}-1)\beta_{2}^{3}(1+\beta_{1})^{3}-e_{3}^{2}[\alpha_{2}( \alpha_{1}-1)+\beta_{1}(1-\alpha_{2})](1+\beta_{2})^{3}=(\alpha_{1}-1)\beta_{ 2}^{3}(1+\beta_{1})(1-\beta_{1})^{2}\\ -[\alpha_{2}(\alpha_{1}-1)+\beta_{1}(1-\alpha_{2})]\times(1+\beta_{2})(1- \beta_{2})^{2}-2\beta_{1}\beta_{2},\end{array}  \tag{5}\]  
\[ \begin{array}{c}S_{II}=0\\ \\ e_{1}^{2}=1+\frac{1}{\beta_{1}^{3}\beta_{2}^{2}}[\alpha_{1}\alpha_{2}(1+\beta _{2})(1-\beta_{2})^{2}-\alpha_{1}\beta_{2}^{3}(1+\beta_{1})(1-\beta_{1})^{2}\\ +e_{2}^{2}\alpha_{1}\beta_{2}^{3}(1+\beta_{1})^{3}-e_{3}^{2}\alpha_{1}\alpha_{ 2}(1+\beta_{2})^{3}],\end{array}  \tag{6}\]  
Figure 2: Schematic diagram of the initial structure of the three-mirror system.