# A Miniaturized Design for a Terahertz Tri-Mirror CATR with High QZ Characteristics
## 2 Design of a Miniaturized Tri-Mirror CATR
### Cross-Polarization Cancellation Method and Geometric Configuration  
For a tri-mirror CATR, the geometric positions of all the mirrors have a significant influence on the system's cross-polarization. Therefore, to realize low cross-polarization for a miniaturized tri-mirror CATR, it is essential to first determine the geometric layout of the system under low cross-polarization constraints and calculate precisely each reflector's center coordinates. As illustrated in Figure 1, the developed tri-mirror system comprises a feed, two small mirrors, and a paraboloid primary mirror. Specifically, mirror 1 is configured with confocal foci at \(F_{0}\) and \(F_{1}\), while mirror 2 shares confocal foci at \(F_{1}\) and \(F_{2}\), and the paraboloid primary mirror has its focal point at \(F_{2}\). The equivalent focal lengths of mirror 1, mirror 2, and the primary mirror are denoted \(f_{1}\),\(f_{2}\), and \(f_{3}\), respectively. In this system, cross-polarization cancelation conditions based on beam mode expansion theory have been adopted for the system's structural configuration design and the calculation of optical center coordinates in each mirror. Figure 1: Schematic diagram and parameter variables of the tri-mirror system. In order to enable the tri-mirror system to achieve a low cross-polarization structural layout, according to the definition of the equivalent focal length of the mirrors in multi-mirror systems and the derivation of the low cross-polarization formula in [29], the parameter variables of the designed tri-mirror system illustrated in Figure 1 should satisfy the following equations:  
\[\frac{s_{2}\text{tan}(\sigma_{3}/2)}{f_{3}\cdot s_{1}\text{tan}( \sigma_{1}/2)}=\frac{1}{s_{0}}+\frac{1}{l} \tag{1}\] \[\frac{1/s_{1}+1/s_{2}}{\{f_{3}\text{tan}(\sigma_{2}/2)/s_{1}\text {tan}(\sigma_{3}/2)\}-1}=\frac{1}{f_{3}-s_{2}}-\frac{1}{s_{1}-l}\] (2) \[f_{2}=\frac{\{f_{3}\text{tan}(\sigma_{2}/2)/s_{2}\text{tan}( \sigma_{3}/2)\}-1}{1/s_{1}+1/s_{2}} \tag{3}\]  
where variable \(l\) denotes the distance between the optical center of mirror 1 and the confocal focus, \(F_{1}\). \(S_{l}\) (\(i\) = 1, 2, 3) denotes the distance of the central beam in each of the three propagation segments as rays emitted from the feed sequentially pass through mirror 1, mirror 2, and the primary mirror. \(\sigma_{i}\) (\(i\) = 1, 2, 3) is denoted as the angle formed by the incident central beam and the reflected central beam of each mirror. \(\alpha\) is the angle formed between the propagation direction of the central reflected beam of mirror 1 and the horizontal axis direction, with \(\alpha\) defined as positive in the counterclockwise direction. \(\theta_{0}\) is the angle formed between the radiation direction of the central beam of the feed and the horizontal axis direction. All the above variables are marked in the system depicted in Figure 1.  
Next, the value relationships of the above-mentioned structural parameter variables will be analyzed and solved, and the geometric layout of the tri-mirror system will be determined. Guided by the principles of miniaturization and practicality, the primary mirror and mirror 1 in the tri-mirror system will be assumed to both be concave mirrors. According to the cross-polarization cancelation condition based on the beam mode analysis method and the system's structural definition in [29], it can be seen that \(f_{1}>0,f_{3}>0\). Additionally, when the central ray between mirror 1 and mirror 2 lies above the horizontal axis passing through the optical center of mirror 1 (i.e., \(\alpha>0\)), the following conditions hold: \(\sigma_{2}>\sigma_{3},f_{3}>S_{2}\). Thus, the following relationship can be derived:  
\[\frac{f_{3}}{s_{2}}>\frac{\text{tan}(\sigma_{3}/2)}{\text{tan}( \sigma_{2}/2)} \tag{4}\]  
Then, substituting Equation (4) into Equation (3) yields \(f_{2}<0\); as stated in [29], mirror 2 is a convex mirror.