# Correlation Plenoptic Imaging: An Overview
## 4 A Different Architecture of CPI with Chaotic Light  
We are now going to discuss the possibility of performing correlation plenoptic imaging in a setup that also enables standard imaging, namely, where the image of the object is available from ordinary intensity measurements. The advantage is the possibility to directly monitor the object while plenoptic imaging is performed. The proposed setup is represented in Figure 11; it is characterized by the same components as the scheme discussed in Section 2, but the light source and the lens now play opposite roles. Actually, the lens \(L_{b}\) now focuses the object on detector \(\mathrm{D}_{b}\), while the chaotic source enables for reproducing the ghost image of the lens by intensity correlation measurements [69]. For the ghost image of the lens to be focused on \(\mathrm{D}_{b}\), we set  
\[z_{a}=z_{b}+S_{1}. \tag{17}\]  
Figure 10: Experimental setup for the proof-of-principle demonstration of correlation plenoptic imaging with chaotic light. Adapted with the permission from [42], copyright American Physical Society, 2017. The spatial (\(\mathrm{D}_{a}\)) and the angular (\(\mathrm{D}_{b}\)) sensors are part of the same scientific complementary metal-oxide semiconductor camera (sCMOS). The additional lens \(L_{a}\) reproduces the “ghost image plane” on sensor \(\mathrm{D}_{a}\). The choice of imaging the lens \(L_{b}\) is motivated by the need to gain directional information on light propagating from the object to \(\mathrm{D}_{b}\). In fact, in the present scenario, \(L_{b}\) is the focusing element, responsible for reproducing the image of the object of interest. The object is focused on \(\mathrm{D}_{b}\) when the lens-to-sensor distance is \(S_{2}=S_{2}^{f}\), with  
\[\frac{1}{S_{1}}+\frac{1}{S_{2}^{f}}=\frac{1}{f}; \tag{18}\]  
however, since we are interested in demonstrating the refocusing power of this scheme, we shall not fix the value of \(S_{2}\). The correlation function is determined by an integral involving the source intensity profile \(\mathcal{S}(\mathbf{\rho}_{s})\), the object aperture function \(A(\mathbf{\rho}_{o})\) and the lens pupil function \(P(\mathbf{\rho}_{\ell})\). However, in line with realistic experimental conditions, we shall assume that the source is large enough not to affect propagation of light in arm \(b\) of the setup; in this case, the value of \(\Gamma\) is determined with good approximation by the object features and the lens aperture, namely (up to irrelevant constants),  
\[\Gamma_{(S_{1},S_{2})}(\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\left|\int d\mathbf{\rho}_{o}d \mathbf{\rho}_{\ell}A(\mathbf{\rho}_{o})P(\mathbf{\rho}_{\ell})e^{i\frac{\omega_{0}}{c} \psi(\mathbf{\rho}_{o},\mathbf{\rho}_{\ell};\mathbf{\rho}_{a},\mathbf{\rho}_{b})}\right|^{2}, \tag{19}\]  
where  
\[\psi(\mathbf{\rho}_{o},\mathbf{\rho}_{\ell};\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\left(\frac{ 1}{S_{2}}-\frac{1}{S_{2}^{f}}\right)\frac{\mathbf{\rho}_{\ell}^{2}}{2}-\left(\frac {\mathbf{\rho}_{o}}{S_{1}}+\frac{\mathbf{\rho}_{b}}{S_{2}}\right)\cdot\mathbf{\rho}_{\ell} +\frac{\mathbf{\rho}_{o}\cdot\mathbf{\rho}_{a}}{S_{1}}. \tag{20}\]  
With respect to the result obtained for the scheme of Figure 1, the source intensity profile is replaced here by the pupil function of the lens, and the focusing condition \(z_{a}=z_{b}\) is replaced by \(S_{2}=S_{2}^{f}\). Figure 11: Alternative scheme to perform CPI with chaotic light, while monitoring the object by standard imaging. The lens \(L_{b}\) reproduces the image of the object on detector \(\mathrm{D}_{b}\), thus playing the role of the focusing element. The (ghost) image of this lens is retrieved on \(\mathrm{D}_{a}\) by correlation measurements. Information on the direction of light from the object plane to the lens, as given by the correlation of intensity fluctuations, enables refocusing objects placed out of the conjugate plane of the object, namely at \(S_{2}\
eq S_{2}^{f}\).