# A Miniaturized Design for a Terahertz Tri-Mirror CATR with High QZ Characteristics
## 2 Design of a Miniaturized Tri-Mirror CATR
### Synthesis of Shaped Mirrors and System Design  
After determining the geometric layout of the tri-mirror CATR and the optical center coordinates of each mirror, dynamic ray tracing based on GO and the dual-paraboloid expansion method in [31] are employed to synthesize the two shaped mirrors of the system. The shaped mirror synthesis process adheres to the principles of energy conservation and equal optical path. Dividing the spherical beam radiated from the feed source into an infinite number of sub-beams establishes a one-to-one correspondence with countless discrete target points in the desired exit QZ field of radius, \(r\), while redistributing energy. The entire beam-tracing process employs the double paraboloid expansion method, starting with the central beam emitted from the feed source, which sequentially expands outward along the optical central points of each mirror and undergoes beam rearrangement until the final sub-beam is traced. The innumerable sub-beams intersect the spatial surfaces containing the optical centers of the two shaped mirrors, forming densely packed and smoothly connected discrete points that collectively compose the shaped mirror surfaces. Following the design procedure described above, a mapping function (\(x\), \(y\)) = \(F(\theta\), \(\varphi\)) between the beam with the output direction (\(\theta\), \(\varphi\)) emitted from the feed and the position (\(x\), \(y\)) where the desired output field is established using the dynamic ray tracing method [31]. This process is programmed and numerically computed in the numerical software MATLAB R2018b to generate numerous discrete data points representing the two shaped mirrors. The data are then imported into the electromagnetic simulation software GRASP-10, which is based on physical optics and the physical theory of diffraction, for modeling and simulation. The combination of these two shaped mirrors and the paraboloid primary mirror can significantly improve the QZ characteristics of the system. Furthermore, analysis of the designed feed and the desired output QZ field reveals that they are two critical factors influencing the system's QZ characteristics. For the horn feed, its normalized far-field is represented by  
\[G(\theta)=10^{\frac{7_{1}}{20}(\frac{\theta_{1}}{r_{1}})^{2}} \tag{5}\]  
where \(\theta\) is the angle of the output field relative to the boresight direction of the feed, and \(T_{1}\) represents the edge taper of the feed at a specific angle, \(\theta_{1}\). For a fixed feed flare-angle, a larger edge taper results in a steeper drop-off in the far-field distribution, leading to more concentrated beam energy radiated by the feed and reduced diffraction effects in the system. However, higher edge taper values increase the fabrication complexity of the feed. Therefore, in high-performance CATR designs, the edge taper can be treated as an optimization variable to balance performance and manufacturability. For the expected output QZ field, its field strength distribution can be expressed as  
\[E(r)=\left\{ \begin{array}{l}1\,,\quad\quad r_{\rm q}\geq r\geq 0\\ 10^{\frac{7_{0}}{20}(\frac{r-r_{\rm q}}{\max-r_{\rm q}})^{2}},\quad r_{\rm max} \geq r>r_{\rm q}\end{array} \right. \tag{6}\]  
where \(r\) is the radius of the output field; \(r_{\rm q}\) is the expected QZ radius; \(r_{\rm max}\) represents the radius of the system's primary mirror; and \(T_{0}\) is the expected edge taper of the output field  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Parameter** & \(\theta_{0}\) (\({}^{\circ}\)) & \(\alpha\) (\({}^{\circ}\)) & \(\sigma_{1}\) (\({}^{\circ}\)) & \(\sigma_{2}\) (\({}^{\circ}\)) & \(\sigma_{3}\) (\({}^{\circ}\)) & \(S_{0}\) (m) & \(S_{1}\) (m) & \(S_{2}\) (m) & \(I\) (m) & \(f_{3}\) (m) \\at \(r_{\rm max}\). The expected output QZ field features a flat distribution in the center region and Gaussian steep drop-offs at both edges.