# Fast Computational Ghost Imaging with Laser Array Modulation
## 2 Computational Ghost Imaging with Laser Array  
Figure 1 shows the optical layout of the proposed system. The system consists of an array of laser diodes, an SLM for random phase modulation, a convex lens, and a photo-detector. For the laser array, we will consider vertical-cavity surface-emitting lasers (VCSELs). Note that all elemental emitters have same characteristics such as spectrum, output power, and so on. First, such arrays are suitable for integration on silicon substrates. Second, with VCSELs, it is easy to implement a two-dimensional array of emitters. Finally, VCSELs can be switched on a scale of several GHz; this broad bandwidth is an important feature of the proposed method. In the optical system, each element on the laser array is emitted one by one per a measurement.  
The procedure of the measurement in the proposed method is as follows. First, one source emits light; then, the speckle patterns generated by scattering on the SLM illuminate the target object. Here, the intensity distributions of the speckle patterns are defined as \(I_{1}(x,y)\). The optical signals transmitted through the target are collected at the photo-detector by the condenser lens, with the intensity of the detected signal defined as \(B_{1}\). By iteratively changing the emitting position in the laser array, a set of measured data \(B_{i}\) (\(i=1,\,2,\,...,\,n\)) is obtained.  
Figure 1: Schematic of an optical system for computational ghost imaging with laser array modulation.  
According to [13; 14], the intensity transmittance of the target \(T(x,y)\) is obtained as follows:  
\[T(x,y)=\frac{1}{n}\sum\
olimits_{i=1}^{n}(B_{i}-<B>)I_{i}(x,y) \tag{1}\]  
\(I_{i}(x,y)\) is derived by numerical preprocessing as in the conventional method. In Equation (1), \(<\)\(B>\) indicates the average of \(B_{i}\) expressed as  
\[<B>\ =\frac{1}{n}\sum\
olimits_{i=1}^{n}B_{i} \tag{2}\]  
Equation (1) is based on the second order statistical properties of the speckle intensity. This equation is derived from the intensity autocorrelation function, with the details of the derivation described in Ref. [4].  
CGI generally requires more than thousands of measurements for image reconstruction. In conventional CGI, the illumination speckle patterns are changed by refreshing the random phase modulation on the SLM. A liquid crystal on silicon (LCOS) device is often used for the SLM. The refresh rate of the device is much smaller than 1 kHz. Therefore, conventional CGI is considered to be able to work at the video rate (30 Hz). The bottleneck for the measurement speed is the rate of changing the illumination patterns.  
In contrast, in our proposed method, the random phase patterns on the SLM are unchanged during the measurement. The speckle patterns in the target plane depend on the distributions of the optical signals on the SLM. These distributions are determined by the emitting position in the VCSEL array. Therefore, a set of speckle patterns can be obtained by controlling the laser array. The broad bandwidth of the VCSELs should allow a dramatic reduction in the CGI measurement time. Let us consider imaging with the proposed CGI in 30 frame per second. Also, it is assumed that a reconstructed image is obtained by 10,000 times measurements. In this case, each emitter on the laser array should be operated in 300 kHz. Also, our method can employ a random diffuser as alternative to the SLM. In this case, a set of speckle intensity distributions \(I_{i}(x,y)\) should be acquired in premeasurement. Although this implementation is not CGI, it is considered to be useful for simple setup and improvement of robustness for GI.