# Beam Scanning and Capture of Micro Laser Communication Terminal Based on MEMS Microm mirrors
## 2 MEMS Micromirror Optical Scanning Model
\begin{bmatrix}-\sin 2\beta&0&-\sin\beta\\ 0&1&0\\ -\cos 2\beta&0&-\cos\beta\end{bmatrix}\begin{bmatrix}-\sin 2\beta\\ 0\\ -\cos 2\beta\end{bmatrix} = \begin{bmatrix}0\\ 0\\ 1\end{bmatrix} . \tag{9}\]  
Equation (9) confirms that the intersection point of \(\overset{\rightarrow}{\mathbf{A^{\prime}}}\) and the \(X^{\prime}O^{\prime}Y^{\prime}\) plane in the initial state is the origin \(O^{\prime}\), and the outgoing light overlaps with the \(Z^{\prime}\) axis. When the MEMS mirror undergoes vibrations and deflections in the 2D space, the aforementioned theory will be used to calculate the intersection coordinates of the projection line and the \(X^{\prime}O^{\prime}Y^{\prime}\) plane as well as the deflection angle of the \(Z^{\prime}\) axis. Assuming that the horizontal deflection angle of the MEMS micromirror is \(\alpha_{x}\), the vertical deflection angle is \(\alpha_{y}\). This is equivalent to the \(XOY\) plane in Figure 1: first rotating \(\alpha_{y}\) counterclockwise with the \(y\)-axis as the axis of rotation and then rotating \(\alpha_{x}\) clockwise with the \(x\)-axis as the axis of rotation. The new normal unit vector corresponding to the reflective surface of the MEMS micromirror can be expressed as Equation (10). \[\overset{\rightarrow}{\boldsymbol{N}}= \begin{bmatrix}N_{x}\\ N_{y}\\ N_{z}\end{bmatrix} = \begin{bmatrix}\sin(\beta+\alpha_{x})\cos\alpha_{y}\\ \sin\alpha_{y}\\ \cos(\beta+\alpha_{x})\cos\alpha_{y}\end{bmatrix} . \tag{10}\]  
After calculation and simplification, the output light in the deflected state of the mirror was obtained:  
\[\boldsymbol{A}^{\prime}=\boldsymbol{A}\boldsymbol{V}= \begin{bmatrix}\cos^{2} \alpha_{y}\sin 2\alpha_{x}-\sin^{2}\alpha_{y}\sin 2\beta\\ \cos(\beta-\alpha_{x})\sin 2\alpha_{y}\\ \cos^{2}\alpha_{y}\cos 2\alpha_{x}-\sin^{2}\alpha_{y}\sin 2\beta\end{bmatrix} . \tag{11}\]  
Regardless of the deviation of the reflecting mirror, the starting point of the output light was always \(\mathcal{O}^{\prime}(0,0,-L)\). Therefore, the linear equation of the output light was  
\[\frac{X^{\prime}}{A_{x}^{\prime}}=\frac{Y^{\prime}}{A_{y}^{\prime}}=\frac{Z^{ \prime}+L}{A_{z}^{\prime}}. \tag{12}\]  
The coordinates of the intersection point between the incident light and the \(X^{\prime}\mathcal{O}^{\prime}Y^{\prime}\) plane was obtained as follows:  
\[\left\{ \begin{aligned} X^{\prime}&=L\frac{\cos^{2} \alpha_{y}\sin 2\alpha_{x}-\sin^{2}\alpha_{y}\sin 2\beta}{\cos^{2}\alpha_{y}\cos 2 \alpha_{x}-\sin^{2}\alpha_{y}\cos 2\beta}\\ Y^{\prime}&=L\frac{\cos(\beta-\alpha_{x})\sin 2\alpha_{y}}{ \cos^{2}\alpha_{y}\cos 2\alpha_{x}-\sin^{2}\alpha_{y}\cos 2\beta}.\end{aligned} \right. \tag{13}\]  
It can be seen from Equation (13) that when \(\alpha_{y}=0\), the MEMS micromirror was deflected only in the horizontal direction, and \(X=L\tan 2\alpha_{x}\) was obtained, indicating that the new reflected line was deflected at an angle of \(2\alpha_{x}\) to the original reflected line. This meant that the optical reflection angle in the horizontal direction was twice the mechanical deflection angle of the micromirror. On the other hand, when \(\alpha_{x}=0\) the relationship was not satisfied in the vertical direction. Since the condition for this conclusion is that the initial incident light was horizontally incident, it lay in the horizontal plane. When \(\alpha_{x}\to 0,\alpha_{y}\to 0\), indicating that the deviation angle of the MEMS mirror was extremely small, we approximated \(\sin\alpha\to 0,\cos\alpha\to 1\), which implied \(Y/X\rightarrow\cos\beta\). The equation showed that when the mechanical deflection angle was small, the ratio of the vertical scan length to the horizontal scan length of the reflector was approximately the cosine of the incident light angle; that is, the scanning effect of the MEMS micromirror was related to the incident angle of the initial incident light.