# Angle of Arrival for the Beam Detection Method of Spatially Distributed Sensor Array
## 2 Methods and Design  
### Detection Principle  
The spatially distributed sensor array is a polyhedral structure, with each face equipped with a sensor that has identical spectral responsivity and photosensitive area. The output of the optical sensor has an angular dependence on the incident light [14], as shown in Figure 1a. The effective area of the detector can be expressed as:  
\[A_{eff}=A\cos\theta \tag{1}\]  
where \(A_{eff}\) is the effective area of the sensor, \(A\) is the area of the sensor, and \(\theta\) is the angle between the incident light and the sensor normal vector.  
The incident light generates photocurrent proportional to the irradiance and the effective area. It can be expressed by the following equation:  
\[U=P\mathcal{R}_{\lambda}R_{L}A_{eff} \tag{2}\]  
where \(U\) is the voltage signal on the load resistor \(R_{L}\), \(\mathcal{R}_{\lambda}\) is the responsivity of the photodetector at a wavelength of \(\lambda\), and \(P\) is the light flux received by the sensor from normal incident light. AOA is calculated by combining the optical power received by each sensor at different positions in space. The position of each sensor in the spatially distributed sensor array is shown in Figure 1b. The matrix \(\mathbf{K}\), composed of the coordinate matrices of the sensor normal vectors, can be expressed as:  
\[\mathbf{K}=\left[\mathbf{n}_{1},\mathbf{n}_{2},\ldots,\mathbf{n}_{i}\ldots,\mathbf{n}_{j}\right] \tag{3}\]  
where \(\mathbf{n}_{i}\) represents the position coordinate vector of each sensor, which is expressed as:  
\[\mathbf{n}_{i}=\left(x_{i},y_{i},z_{i}\right)^{T} \tag{4}\]  
The optical power \(P_{i}\) incident on each sensor is given by:  
\[P_{i}=\int\mathbf{S}\cdot\mathbf{n}_{i}da \tag{5}\]  
where \(\mathbf{S}\) is the energy flux density vector of the incident light, which can be expressed in the Cartesian coordinate system as follows:  
\[\mathbf{S}=S\cdot\mathbf{s}=S\cdot(X,Y,Z)=S\cdot(\cos\beta\text{cos}\gamma,\sin\beta \text{cos}\gamma,\sin\gamma) \tag{6}\]  
where \(S\) is the total energy flux density of the incident light, \(\beta\) is the azimuth angle, and \(\gamma\) is the elevation angle. The output matrix of the spatially distributed sensor array is expressed as:  
\[\mathbf{P}=\left[P_{1}\quad\cdots P_{1}\cdots\quad P_{j}\right]^{T}=S\mathbf{K^{T}} \mathbf{s} \tag{7}\]  
According to Equation (6), there are three parameters to be measured: \(S,\ \beta\), and \(\gamma\). Therefore, at least three sensor output powers with uncorrelated normal vectors should be selected from \(\mathbf{K}\) to calculate the AOA. Matrix \(\mathbf{B}\) is extracted as a block from matrix \(\mathbf{K}\). The  
Figure 1: (**a**) Schematic diagram of the relationship between sensor effective area and incident light. (**b**) Schematic diagram of sensor position and AOA.  
rank of matrix \(B\) satisfies \(r(B)\geq 3\) and its corresponding power output matrix is \(\mathbf{P_{B}}\). The vector \(\mathbf{s}\) can be calculated by:  
\[\mathbf{s}=\frac{1}{S}\Big{(}\mathbf{B}\mathbf{B}^{T}\Big{)}^{-1}\mathbf{B}\mathbf{P_{B}}=(X,Y,Z) \tag{8}\]  
Thus the \(\beta\) and \(\gamma\) for the angle of arrival are given by: [18]:  
\[\begin{array}{c}\beta=\tan^{-1}\frac{\gamma}{X}\\ \gamma=\tan^{-1}\frac{Z}{\sqrt{X^{2}+Y^{2}}}\end{array} \tag{9}\]