# Correlation Plenoptic Imaging: An Overview
## Appendix A Theoretical Background of of CPI with Chaotic Light  
The second-order correlation between the intensities at two points of sensors \(\mathrm{D}_{a}\) and \(\mathrm{D}_{b}\), is described by the Glauber four-point correlation function [83]  
\[G^{(2)}(\mathbf{\rho}_{a},\mathbf{\rho}_{b};t_{a},t_{b})\!=\langle E_{a}^{(-)}(\mathbf{\rho} _{a},t_{a})E_{b}^{(-)}(\mathbf{\rho}_{b},t_{b})E_{b}^{(+)}(\mathbf{\rho}_{b},t_{b})E_{a }^{(+)}(\mathbf{\rho}_{a},t_{a})\rangle, \tag{10}\]  
where \(\mathbf{\rho}_{i}=(x_{i},y_{i})\) is the coordinate on the detector \(\mathrm{D}_{i}\) (with \(i=a,b\)), \(t_{i}\) is the time at which the signal at \(\mathrm{D}_{i}\) is detected, and \(E_{i}^{(\pm)}\) are the components of the electric field, related by \(E^{(+)}=(E^{(-)})^{\dagger}\), with positive and negative frequencies, respectively. We use the scalar approximation for the electric field because, in our setup, light has either a spatial structure independent of polarization or a fixed polarization. In Equation (10), the expectation value is taken over the quantum state of the source (i.e., \(\langle O\rangle=\text{Tr}(\varrho O)\)). The transfer functions \(g_{i}(\mathbf{\rho},\mathbf{k})\) relates the fields at \(\mathbf{\rho}_{i}\) to the modes \(k\) of the field emitted by the source [84], which is  
\[E_{i}^{(+)}(\mathbf{\rho}_{i},t_{i})=C\int d\omega\int d\kappa\,a_{k}e^{-i\omega t _{i}}g_{i}(\mathbf{\rho}_{i},\mathbf{k}). \tag{11}\]  
In the above equation, \(\omega=c|\mathbf{k}|\) is the frequency of the mode \(\mathbf{k}\), \(\kappa\) its transverse momentum, and \(a_{\mathbf{k}}\) is the mode annihilation operator, which, together with the creation operator \(a_{\mathbf{k}^{\prime}}^{\dagger}\), satisfies the canonical commutation relation \([a_{\mathbf{k}},a_{\mathbf{k}^{\prime}}^{\dagger}]=\delta_{\mathbf{k},\mathbf{k}^{\prime}}\); \(C\) is an irrelevant constant. In the paraxial approximation, the longitudinal component \(k_{z}\) of the wave vector is such that \(\omega\simeq ck_{z}\). For stationary and quasi-monochromatic sources, characterized by a central frequency \(\omega_{0}\), the combination of Equations (11) and (10) leads to non-vanishing expectation values \(\langle a_{\mathbf{k}_{1}}^{\dagger}a_{\mathbf{k}_{2}}^{\dagger}a_{\mathbf{k}_{3}}a_{\mathbf{ k}_{4}}\rangle\) only if \(|\mathbf{k}_{g}|\simeq\omega_{0}/c\) for \(i=1,\dots,4\). In this case, the Glauber four-point function in Equation (10) depends only on the time difference \(\tau=t_{a}-t_{b}\). Moreover, the correlation function becomes the product of a time-dependent and a space-dependent part. If the source is chaotic, the correlation function is made of two terms [62]  
\[\langle a_{\mathbf{k}_{1}}^{\dagger}a_{\mathbf{k}_{2}}^{\dagger}a_{\mathbf{k}_{3}}a_\mathbf{{k}_{4}}\rangle\propto\delta(\mathbf{k}_{1}-\mathbf{k}_{4})\delta(\mathbf{k}_{2}-\mathbf{k}_{3 })+\delta(\mathbf{k}_{1}-\mathbf{k}_{3})\delta(\mathbf{k}_{2}-\mathbf{k}_{4}), \tag{12}\]  
which are related to the bosonic symmetrization of two-photon states. Notice that the above result strictly holds for an indefinitely extended and flat chaotic source. If the source is characterized by a nontrivial amplitude profile, we can still keep the result (3) and insert the information on the spatial modulation in the propagators. Hence, if one neglects the time dependence by only considering detection time differences (\(\tau\)) which are much smaller than the source coherence time, the correlation function in Equation (10), as evaluated for a stationary, quasi-monochromatic and chaotic source, reads  
\[G^{(2)}(\mathbf{\rho}_{a},\mathbf{\rho}_{b})=I_{a}(\mathbf{\rho}_{a})I_{b}(\mathbf{\rho}_{b})+ \Gamma(\mathbf{\rho}_{a},\mathbf{\rho}_{b}), \tag{13}\]  
where the first term is the product average of intensities:  
\[I_{i}(\mathbf{\rho}_{i})=\int d\kappa|g_{i}(\mathbf{\rho}_{i},\kappa)|^{2} \tag{14}\]at the pixel located in \(\mathbf{\rho}_{i}\) of the sensor \(\mathrm{D}_{i}\); notice that the frequency dependence has been dropped in the transfer functions \(g_{i}\).