# Correlation Plenoptic Imaging: An Overview
## Appendix A Theoretical Background of of CPI with Chaotic Light
The second term  
\[\Gamma(\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\left|\int d\mathbf{\kappa}g_{a}(\mathbf{\rho}_{a}, \mathbf{\kappa})^{*}g_{b}(\mathbf{\rho}_{b},\mathbf{\kappa})\right|^{2} \tag{10}\]  
represents the nontrivial part of the second-order correlation, which can be used to encode either standard or plenoptic imaging properties. In order to unveil such properties, we first need to compute the transfer functions \(g_{a}\) and \(g_{b}\). To this end, we use the paraxial Gaussian propagator [84]. \[\mathcal{G}(\mathbf{\rho},z;\omega)=G(\mathbf{\rho})_{\left[\frac{\omega}{cz}\right]} h(\omega,z), \tag{11}\]  
with  
\[G(\mathbf{\rho})_{\left[\beta\right]}=\exp\left(\frac{i}{2}\beta\rho^{2}\right), \quad h(\omega,z)=-i\frac{\omega}{2\pi cz}e^{i\frac{\omega}{c}z}, \tag{12}\]  
and treat the source as an emitter of paraxial waves. In arm \(a\), light propagates in free space from the source to the detector \(\mathrm{D}_{a}\). Hence, the corresponding transfer function is  
\[g_{a}(\mathbf{\rho}_{a},\mathbf{\kappa})=h(\omega_{0},z_{a})\int\!d\mathbf{\rho}_{s}f(\bm {\rho}_{s})e^{i\kappa\cdot\mathbf{\rho}_{s}}G(\mathbf{\rho}_{a}-\mathbf{\rho}_{s})_{\left[ \frac{\omega_{0}}{cz_{a}}\right]}=\mathcal{C}_{a}(\mathbf{\rho}_{a},z_{a})\int\!d \mathbf{\rho}_{s}f(\mathbf{\rho}_{s})e^{\left(\kappa-\frac{\omega_{0}}{cz_{a}}\mathbf{ \rho}_{s}\right)\cdot\mathbf{\rho}_{s}}G(\mathbf{\rho}_{s})_{\left[\frac{\omega_{0}}{ z_{a}}\right]}\quad, \tag{13}\]  
where \(f(\mathbf{\rho}_{s})\) is the source amplitude profile, and  
\[\mathcal{C}_{a}(\mathbf{\rho}_{a},z_{a})=h(\omega_{0},z_{a})G(\mathbf{\rho}_{a})_{ \left[\frac{\omega_{0}}{cz_{a}}\right]}. \tag{14}\]  
Computation of the field propagator \(g_{b}\) requires additional integration on both the object and the lens plane, namely:  
\[E_{i}^{(+)}(\mathbf{\rho}_{i},t_{i})=C\int d\omega\int d\kappa\,a_{k}e^{-i\omega t _{i}}g_{i}(\mathbf{\rho}_{i},\mathbf{k}). \tag{11}\]0 \[E_{i}^{(+)}(\mathbf{\rho}_{i},t_{i})=C\int d\omega\int d\kappa\,a_{k}e^{-i\omega t _{i}}g_{i}(\mathbf{\rho}_{i},\mathbf{k}). \tag{11}\]1  
where \(A(\mathbf{\rho}_{o})\) and \(L(\mathbf{\rho}_{\ell})\) are the transmission functions of the object and the lens, respectively. Henceforth, we shall assume that the finite size of the lens \(L_{b}\) is irrelevant for propagation in arm \(b\), and approximate its transmission function with the Gaussian phase shift \(G(\mathbf{\rho}_{\ell})_{\left[-\omega_{0}/cF\right]}\), thus neglecting the pupil function. This assumption is justified by the fact that the spot on the lens is limited by the size of both the source and the transmissive part of the object. If light falls outside the lens pupil, the finite size of \(L_{b}\) must be taken into account. Assuming that the distance from the source to the lens (\(S_{o}\)) and from the lens to the detector (\(S_{i}\)) are conjugate (i.e., \(1/S_{i}+1/S_{o}=1/F\)), the propagator in arm \(b\) reads  
\[E_{i}^{(+)}(\mathbf{\rho}_{i},t_{i})=C\int d\omega\int d\kappa\,a_{k}e^{-i\omega t _{i}}g_{i}(\mathbf{\rho}_{i},\mathbf{k}). \tag{11}\]2  
where \(M=S_{i}/S_{o}\) is the lens magnification and  
\[E_{i}^{(+)}(\mathbf{\rho}_{i},t_{i})=C\int d\omega\int d\kappa\,a_{k}e^{-i\omega t _{i}}g_{i}(\mathbf{\rho}_{i},\mathbf{k}). \tag{11}\]3Given the propagators of Equations (A9) and (A12), it is now straightforward to compute the correlation function of Equation (A4).