# Correlation Plenoptic Imaging: An Overview
## 2 Correlation Plenoptic Imaging with Chaotic Light--First Scheme
### Correlation Functions in CPI  
The correlation between the fluctuations \(\Delta I_{a}\) and \(\Delta I_{b}\) of the intensities \(I_{a}\) and \(I_{b}\), evaluated at the transverse coordinates \(\mathbf{\rho}_{a}\) and \(\mathbf{\rho}_{b}\) defined on the planes of detectors \(\mathrm{D}_{a}\) and \(\mathrm{D}_{b}\), respectively, reads  
\[\Gamma(\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\langle\Delta I_{a}(\mathbf{\rho}_{a})\Delta I _{b}(\mathbf{\rho}_{b})\rangle=\langle I_{a}(\mathbf{\rho}_{a})I_{b}(\mathbf{\rho}_{b}) \rangle-\langle I_{a}(\mathbf{\rho}_{a})I_{b}(\mathbf{\rho}_{b})\rangle; \tag{1}\]  
the symbol \(\langle\dots\rangle\) denotes the expectation value on the state of the source, and can be determined by taking into account the source statistics. For the setup represented in Figure 1, the computation in the case of a chaotic and quasi-monochromatic source with central frequency \(\omega_{0}\), yields (see Appendix A for details)  
\[\Gamma_{(z_{a},z_{b})}(\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\left|\int\!d\mathbf{\rho}_{o} d\mathbf{\rho}_{s}A(\mathbf{\rho}_{0})\mathcal{S}(\mathbf{\rho}_{s})e^{i\frac{\omega_{0}}{c} \varphi(\mathbf{\rho}_{s},\mathbf{\rho}_{a},\mathbf{\rho}_{b})}\right|^{2}, \tag{2}\]  
with  
\[\varphi(\mathbf{\rho}_{o},\mathbf{\rho}_{s};\mathbf{\rho}_{a},\mathbf{\rho}_{b})=\frac{\rho_{ s}^{2}}{2}\left(\frac{1}{z_{b}}-\frac{1}{z_{a}}\right)-\frac{\mathbf{\rho}_{o}}{z_{b}} \cdot\left(\mathbf{\rho}_{s}+\frac{\mathbf{\rho}_{b}}{M}\right)+\frac{\mathbf{\rho}_{s} \cdot\mathbf{\rho}_{a}}{z_{a}}, \tag{3}\]  
where \(\mathbf{\rho}_{o}\) is the transverse coordinate on the plane of the sample, \(\mathbf{\rho}_{s}\) is the transverse coordinate on the source plane, \(c\) is the speed of light, \(\omega_{0}\) is the pump frequency of the laser pump, \(A\) is the aperture function of the object, \(\mathcal{S}\) is the transverse intensity profile of the source and \(M\) the absolute magnification of the image of the source on \(\mathrm{D}_{b}\). The dependence of \(\Gamma\) on the distances \(z_{a}\) and \(z_{b}\) has been explicitly highlighted to enable easily checking whether focused or out-of-focus images are being considered. Integration of \(\Gamma\) over the whole directional sensor \(\mathrm{D}_{b}\) yields an incoherent image of the object, whose point-spread function (PSF) is related with the Fourier transform of the function \(\mathcal{S}(\mathbf{\rho}_{s})e^{i\frac{\omega_{0}}{2}(z_{b}^{-1}-z_{a}^{-1})\rho_ {s}^{2}}\), which we shall indicate by \(\mathcal{\hat{S}}\). The minimal point-spread thus occurs at \(z_{b}=z_{a}\), where \(\mathcal{\hat{S}}\) coincides with the Fourier transform of \(\mathcal{\hat{S}}\); this is the typical PSF of ghost imaging with chaotic light. In particular, the focused ghost image reads  
\[\Sigma^{\mathrm{foc}}(\mathbf{\rho}_{a})=\!\int\!d\mathbf{\rho}_{b}\Gamma_{(z_{a},z_{ a})}(\mathbf{\rho}_{a},\mathbf{\rho}_{b})\propto\!\int\!d\mathbf{\rho}_{o}|A(\mathbf{\rho}_{o})|^{ 2}\left|\mathcal{\hat{S}}\left[\frac{\omega_{0}}{cz_{a}}\left(\mathbf{\rho}_{o}- \mathbf{\rho}_{a}\right)\right]\right|^{2}, \tag{4}\]  
which entails a quasi one-to-one correspondence between points of the object (\(\mathbf{\rho}_{o}\)) and pixels of the sensor \(\mathrm{D}_{a}\) (\(\mathbf{\rho}_{a}\)), with an uncertainty \(\Delta\rho_{a}=\lambda_{0}z_{a}/D_{s}\) defined by the effective diameter \(D_{s}\) of the  
Figure 2: Unfolded setups (Klyshko-like pictures) of ghost imaging (**a**) and correlation plenoptic imaging (**b**) with chaotic light, as referred to the focused case \(z_{a}=z_{b}\). Adapted with the permission from [40], copyright Walter de Gruyter GmbH, 2016. In ghost imaging, a bucket detector collects light transmitted by the object, with neither spatial nor directional resolution. In CPI, the high-resolution detector \(\mathrm{D}_{b}\) enables measuring point-by-point intensity correlations between the two sensors and simultaneously reconstruct both the transmission profile of the object and the propagation direction of light from the source to the object. source.