# Beam Scanning and Capture of Micro Laser Communication Terminal Based on MEMS Microm mirrors
## 2 MEMS Micromirror Optical Scanning Model  
To investigate the relationship between the deflection angle and planar scanning of the 2D MEMS micromirror, a spatial rectangular coordinate system was established (Figure 1). In the initial state, where it did not deflect, the micromirror coincided with the \(XOY\) plane, and \(\overset{\rightarrow}{\boldsymbol{\Lambda}}\) was the normal unit vector of the \(XOY\) plane coincident with the \(Z\) axis. The incident light \(\overset{\rightarrow}{\boldsymbol{A}}\) formed an angle \(\beta\) with the normal of the MEMS reflection plane on the \(XOZ\) plane. The reflected light \(\overset{\rightarrow}{\boldsymbol{A}^{\prime}}\) was reflected by the MEMS reflective plane, entered the \(X^{\prime}O^{\prime}Z^{\prime}\) plane, and was coplanar with it. The distance between \(O\) and \(O^{\prime}\) was L, the unit normal vector was \(\overset{\rightarrow}{\boldsymbol{N}}\), and the output light was \(\overset{\rightarrow}{\boldsymbol{A}^{\prime}}\). The surface of the MEMS reflective mirror continuously changes as the 2D MEMS mirror vibrates in the horizontal and vertical directions, and this relationship can be derived using the method of matrix optics [12; 14]. The vector form of its reflection law can be expressed in matrix optics by Equation (1). \[\overset{\rightarrow}{\boldsymbol{A}^{\prime}}=\overset{\rightarrow}{ \boldsymbol{A}}-2(\overset{\rightarrow}{\boldsymbol{A}}\cdot\overset{ \rightarrow}{\boldsymbol{N}})\cdot\overset{\rightarrow}{\boldsymbol{N}} \tag{1}\]  
Figure 1: Optical scanning model of 2D MEMS micromirror. where \(\overset{\rightarrow}{\mathbf{A}}\) is the incident light unit vector; \(\overset{\rightarrow}{\mathbf{N}}\) is the normal direction unit vector; and \(\overset{\rightarrow}{\mathbf{A^{\prime}}}\) is the outgoing light unit vector. Transforming the vector form of the reflection law in Equation (1) into a matrix expression, establishes a 3D rectangular coordinate system \(x\), \(y\), \(z\) where \(i,j\), \(k\) represent the directions of the three-dimensional components in the vector analysis, respectively. \[\overset{\rightarrow}{\mathbf{A}}=A_{x}\overset{\rightarrow}{i}+A_{y}\overset{ \rightarrow}{j}+A_{z}\overset{\rightarrow}{k}. \tag{2}\]  
\[\overset{\rightarrow}{\mathbf{A^{\prime}}}=A_{x}^{\prime}\overset{\rightarrow} {i}+A_{y}^{\prime}\overset{\rightarrow}{j}+A_{z}^{\prime}\overset{ \rightarrow}{k}. \tag{3}\]  
\[\overset{\rightarrow}{\mathbf{N}}=N_{x}\overset{\rightarrow}{i}+N_{y}\overset{ \rightarrow}{j}+N_{z}\overset{\rightarrow}{k}. \tag{4}\]  
The outgoing light \(\overset{\rightarrow}{\mathbf{A^{\prime}}}\), the incident light \(\mathbf{A}\) and the reflection matrix \(\mathbf{V}\) can be expressed as Equation (5). \[\mathbf{A^{\prime}}=\mathbf{A}\mathbf{V}. \tag{5}\]  
According to the above description, the expression of the reflection matrix can be obtained as Equation (6). \[\mathbf{V}= \begin{bmatrix}1-2N_{x}^{2}&-2N_{y}N_{x}&-2N_{z}N_{x}\\ -2N_{x}N_{y}&1-2N_{y}^{2}&-2N_{z}N_{y}\\ -2N_{x}N_{z}&-2N_{y}N_{z}&1-2N_{z}^{2}\end{bmatrix} . \tag{6}\]  
The expression of the reflection matrix \(\mathbf{V}\) is only related to the component of the normal unit vector \(\overset{\rightarrow}{\mathbf{N}}\) of the reflection surface in each direction, and the reflection matrix is determined by the spatial position of the reflection surface. When the MEMS micromirror did not vibrate in the initial state using \(X^{\prime}Y^{\prime}Z^{\prime}\) as the principal coordinate system, the following relationship was obtained:  
\[\overset{\rightarrow}{\mathbf{A}}= \begin{bmatrix}A_{x}\\ A_{y}\\ A_{z}\end{bmatrix} = \begin{bmatrix}-\sin 2\beta\\ 0\\ -\cos 2\beta\end{bmatrix} . \tag{7}\]  
\[\overset{\rightarrow}{\mathbf{N}}= \begin{bmatrix}N_{x}\\ N_{y}\\ N_{z}\end{bmatrix} = \begin{bmatrix}-\sin\beta\\ 0\\ -\cos\beta\end{bmatrix} . \tag{8}\]  
Thus, the reflection matrix in the initial state can be obtained as Equation (9). \[\mathbf{A^{\prime}}=\mathbf{A}\mathbf{V}=