## 2 System Working Principle
### Working Principle of the APT System  
In airborne laser communication systems, piezoelectric ceramic or electromagnetically driven mirrors are often used to achieve high-precision alignment and tracking, but their structures are complex and difficult to process, and special control boxes are required for driving. To realize the miniaturization and light weight of the airborne end alignment system, a two-dimensional mirror driven by a stepping motor is used as the main actuator of the airborne end in this paper. The structure of the two-dimensional mirror is shown in Figure 4, which consists of two two-phase hybrid stepping motors, an optical mirror, and an optical precision turntable. By controlling the rotation of the mirror in the two directions of pitch and azimuth, a large-scale and high-frequency dynamic adjustment can be realized, and the rotation of 0-360\({}^{\circ}\) and 0-160\({}^{\circ}\) can be realized in the pitch direction and the azimuth direction, respectively. At the same time, to make the angle of each step of the motor smaller, a subdivision driver is used to subdivide the basic step angle of the stepping motor to achieve higher alignment accuracy. Figure 4: Structure diagram of a two-dimensional mirror. Figure 3: Flow chart of APT system for airborne laser communication. **(a)** Transmitting terminal; **(b)** Airborne terminal. After the emitted beam covers the surface of the airborne end, the scanning capture procedure is performed by the two-dimensional mirror. The two-dimensional mirror can realize the azimuth movement of \(-90\)-\(90^{\circ}\) and the pitch movement of 0-\(360^{\circ}\), and the beam capture process is completed by raster scanning. The reflection principle diagram of the two-dimensional mirror beam is shown in Figure 5. \(A\) is the incident light vector, \(A^{\prime}\) is the beam vector reflected by the mirror, \(N\) is the normal vector of the mirror, \(\theta_{1}\) and \(\theta_{2}\) are the angles of deflection of the two-dimensional mirror around axes \(C_{1}\) and \(C_{2}\), respectively, and \(\alpha\) is the incident angle. According to the reflection theorem of geometric optics, the outgoing ray \(A^{\prime}\) and the normal vector \(N\) can be expressed as [17]:  
\[A^{\prime}=A-2(A\cdot N)N \tag{1}\]  
Let the components of \(A\), \(N\), and \(A^{\prime}\) in the \(XYZ\) coordinate system be brought into Equation (1) to obtain:  
\[ \begin{cases}A^{\prime}_{x}=(1-2N_{x}^{2})A_{x}-2N_{x}N_{y}A_{y}-2N_{x}N_{x}A_ {z}\\ A^{\prime}_{y}=-2N_{x}N_{y}A_{x}+(1-2N_{y}^{2})A_{y}-2N_{y}N_{z}A_{z}\\ A^{\prime}_{z}=-2N_{x}N_{z}A_{x}-2N_{y}N_{z}A_{y}+(1-2N_{z}^{2})A_{z}\end{cases}  \tag{2}\]  
According to Formula (2), reflection is a linear transformation, so Formula (1) can be written as follows:  
\[A^{\prime}=RA \tag{3}\]  
\(R\) is the reflection matrix, which can be expressed as:  
\[R= \begin{bmatrix}1-2N_{x}^{2}&-2N_{x}N_{y}&-2N_{x}N_{z}\\ -2N_{y}N_{x}&1-2N_{y}^{2}&-2N_{y}N_{z}\\ -2N_{z}N_{x}&-2N_{z}N_{y}&1-2N_{z}^{2}\end{bmatrix}  \tag{4}\]  
In Formula (4), \(N_{x}\), \(N_{y}\), and \(N_{z}\) are the projections of the normal vector of the mirror in its reference coordinates. Assuming that the distance between the reflector and the four-quadrant detector is \(S\), the coordinates of the spot on the plane of the four-quadrant detector can be expressed as [15]:  
Figure 5: Schematic diagram of two-dimensional mirror beam reflection. \[\left\{ \begin{array}{l}D_{x}=\lambda_{x}\dfrac{(V_{A}+V_{B})-(V_{C}+V_{D})}{V_{A} +V_{B}+V_{C}+V_{D}}=\lambda_{x}\dfrac{(I_{A}+I_{B})-(I_{C}+I_{D})}{I_{A}+I_{B}+I _{C}+I_{D}}=\lambda_{x}\dfrac{(S_{A}+S_{B})-(S_{C}+S_{D})}{S_{A}+S_{B}+S_{C}+S_ {D}}\\ D_{y}=\lambda_{y}\dfrac{(V_{A}+V_{C})-(V_{B}+V_{D})}{V_{A}+V_{B}+V_{C}+V_{D}}= \lambda_{y}\dfrac{(I_{A}+I_{C})-(I_{B}+I_{D})}{I_{A}+I_{B}+I_{C}+I_{D}}=\lambda _{y}\dfrac{(S_{A}+S_{C})-(S_{B}+S_{D})}{S_{A}+S_{B}+S_{C}+S_{D}}\end{array} \right.