# Fast Computational Ghost Imaging with Laser Array Modulation
## 3 Numerical Verification  
First, the proposed method is verified numerically. The scalar diffraction theory [20] is used for the numerical analysis. Figure 2 shows a diagram explaining the simulation models for the laser array and the SLM, respectively. First, it is assumed that a Gaussian beam is emitted by each light source, with the 1/\(e^{2}\) full width of the beam defined as \(\Delta b\). Then, the optical fields arriving at the SLM plane from the light source are derived according to the Gaussian beam propagation [21]. After the phase modulation in the SLM plane, the optical signals are scattered and propagated to the target plane. To simulate this propagation of the scattered signals, we use the wave field tool [22]. This tool is based on the scalar diffraction theory with angular spectrum [21]. In the analysis, the intensity distributions are denoted by \(I_{i}(x,y)\). \(B_{i}\) is calculated as follows  
\[B_{i}=\sum\
olimits_{x=0}^{M_{m-1}}\sum\
olimits_{y=0}^{M_{n-1}}T(x,y)I_{i}( x,y) \tag{3}\]Finally, a reconstructed image is obtained using Equation (1). The simulation parameters are as follows. In the laser array, the number of lasers and the pitch are defined as \(M_{e}\times N_{e}\) and \(\Delta e\), respectively. The number of pixels and the pixel pitch of the SLM are \(M_{m}\times N_{m}\) and \(\Delta m\). The distance between the laser array and the SLM and that between the SLM and the target plane are also important parameters. The former and the latter are defined as \(L_{1}\) and \(L_{2}\), respectively. The values of the parameters used in the analysis are summarized in Table 1. Also, beam profiles of each element on the VCSEL array is assumed to be gaussian for with the beam waist of 8 \(\upmu\)m.  
Figure 3c depicts the result obtained by the proposed method. \(L_{1}\) and \(L_{2}\) are set to 40 mm and 2000 mm, respectively. The structure of the target roughly appears in the reconstructed image, which shown that the modulation of the laser array is useful for the speckle pattern generation in CGL. However, the image is blurred. Then, when \(L_{2}\) is modified to 400 mm, the averaged speckle size of \(I_{i}(x,y)\) becomes smaller. The reconstructed image obtained with this modification is shown in Figure 3d. In comparison to Figure 3c, the resolution of the reconstructed image is improved by this parameter modification. The reconstructed images in Figure 3b,d are similar. This is because the values of \(L_{1}\) and \(L_{2}\) are set such that the average speckle size of the illumination is the same as in Figure 3b. The shorter \(L_{2}\), the higher is the resolution of the image. On the other hand, the longer \(L_{2}\), the better is the contrast of the reconstructed images. These characteristics are discussed from the theoretical perspective in Ref. [23], including the relation between the contrast and the speckle size. Accordingly, the contrast is improved for larger speckle size. From Ref. [24], the speckle size \(\delta\) is given by the following equation:  
\[\delta=0.9\lambda\frac{L_{2}}{D} \tag{4}\]  
In Equation (4), \(D\) indicates the diameter of the optical distributions on the SLM. Note that \(D\) depends on the distance from the laser array to the SLM (\(L_{1}\)) and \(\Delta b\). In the verification, both parameters are constant values. Therefore, the speckle size of \(I_{i}(x,y)\) depends only on \(L_{2}\). As well known, the speckle size of the illumination patterns affects the resolution of the image reconstruction in CGL. From the reconstructed images shown in Figure 3c,d, it can be seen that the characteristics of the obtained results are consistent with the theoretical analysis. \begin{table}
\begin{tabular}{c c} \hline \(\Delta b\) & 20 \(\upmu\)m \\ \(M_{e}\times N_{e}\) & 100 \(\times\) 100 \\ \(\Delta e\) & 80 \(\upmu\)m \\ \(M_{m}\times N_{m}\) & 1024 \(\times\) 1024 \\ \(\Delta m\) & 8 \(\upmu\)m \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters for numerical analysis.