# Correlation Plenoptic Imaging: An Overview
## 2 Correlation Plenoptic Imaging with Chaotic Light--First Scheme
### Point-Spread Function and Plenoptic Properties  
To better understand the physical meaning of the point-spread-function of CPI, and its consequences on the DOF enhancement, let us consider a light source with a Gaussian intensity profile  
\[\mathcal{S}(\mathbf{\rho}_{s})=\frac{1}{2\pi\sigma^{2}}\exp\!\left(-\,\frac{\mathbf{ \rho}_{s}^{2}}{2\sigma^{2}}\right). \tag{5}\]  
In this case, the _coherent_ PSF in Equation (2) reduces to  
\[\mathcal{C}(\mathbf{\rho}_{o}-\alpha\,\mathbf{\rho}_{a})=\!\int\!d\mathbf{\rho}_{s} \mathcal{S}(\mathbf{\rho}_{s})e^{\frac{\omega_{0}}{c_{2b}}\frac{\mathbf{\rho}_{s}^{2} }{2}(1-\alpha)}e^{-\frac{i\omega_{0}}{c_{2b}}(\mathbf{\rho}_{o}-\alpha\mathbf{\rho}_{a })\mathbf{\rho}_{s}}\propto\exp\left(-\frac{1}{2}\!\left(\frac{\omega_{0}\sigma}{ cz_{b}}\right)^{2}\!\frac{|\mathbf{\rho}_{0}-\alpha\mathbf{\rho}_{a}|^{2}}{1-\frac{i \omega_{0}\sigma^{2}}{cz_{b}}(1-\alpha)}\right), \tag{6}\]  
with \(\alpha=z_{b}/z_{a}\). The PSF of the incoherent image, as obtained by integrating Equation (2) over \(\mathbf{\rho}_{b}\), comes out to be the square modulus of the result in Equation (6), namely,  
\[\mathcal{J}(\mathbf{\rho}_{o}-\alpha\,\mathbf{\rho}_{a})\propto\exp\left(-\left(\frac {\omega_{0}\sigma}{cz_{b}}\right)^{2}\!\frac{|\mathbf{\rho}_{0}-\alpha\mathbf{\rho}_{ a}|^{2}}{1+\left(\frac{\omega_{0}\sigma^{2}}{cz_{b}}(1-\alpha)\right)^{2}} \right). \tag{7}\]  
The physics behind the wider depth of field of CPI with respect to standard ghost imaging comes out from the comparison of Equations (6) and (7). In fact, in geometrical optics (\(\omega_{0}\rightarrow\infty\)), the variance of the incoherent PSF of Equation (7), typical of ghost imaging, approaches \(\sigma\,|1-\alpha|\) and is thus independent of frequency. In the same regime, the coherent PSF of Equation (6), associated with correlation plenoptic imaging, reduces to an imaginary quadratic exponential, with typical width \(\sqrt{2\pi\lambda_{0}z_{b}|1-\alpha|}\). Therefore, the width of the coherent PSF vanishes in the geometrical optics approximation. Most important, the dependence of the two PSFs on defocusing (\(|1-\alpha|\)) is such that the (imaginary) width of the coherent PSF scales much slower than the width of the incoherent PSF. This is the reason behind the wider DOF of (coherent) correlation plenoptic imaging, as compared to incoherent imaging, whether standard or ghost. Now, to unveil the plenoptic properties encoded in \(\Gamma_{(z_{a}z_{b})}\), it is worth considering the limit of Equation (2) in the short-wavelength regime, when the integral is dominated by the stationary points of the phase. In particular, stationarity with respect to \(\mathbf{\rho}_{o}\) yields  
\[\mathbf{\rho}_{s}+\frac{\mathbf{\rho}_{b}}{M}=0, \tag{8}\]  
namely, the geometrical correspondence between points of the source and points of the sensor D\({}_{b}\), with magnification \(M\) and inversion of the image, as already discussed. The stationarity condition with respect to \(\mathbf{\rho}_{s}\) yields a less trivial result, that, combined with Equation (8), identifies the object point \(\mathbf{\rho}_{o}\) that provides the dominant contribution to the integral, at the given detection positions \(\mathbf{\rho}_{a}\) and \(\mathbf{\rho}_{b}\), which is:  
\[\mathbf{\rho}_{o}=\frac{z_{b}}{z_{a}}\mathbf{\rho}_{a}-\frac{\mathbf{\rho}_{b}}{M}\left(1- \frac{z_{b}}{z_{a}}\right). \tag{9}\]Hence, in the geometrical-optics limit, the intensity correlation function reads  
\[\Gamma_{(z_{a},z_{b})}(\mathbf{\rho}_{a},\mathbf{\rho}_{b})\sim\mathcal{S}\Big{(}-\frac{ \mathbf{\rho}_{b}}{M}\Big{)}^{2}\Big{|}A\left[\frac{z_{b}}{z_{a}}\mathbf{\rho}_{a}-\frac {\mathbf{\rho}_{b}}{M}\left(1-\frac{z_{b}}{z_{a}}\right)\right]\Big{|}^{2}\,, \tag{10}\]  
namely, it reproduces the magnified and displaced image of the object (A), multiplied by the source intensity profile, as shown in Figure 3a. When \(z_{b}\
eq z_{a}\), integration over \(\rho_{b}\) blurres the image of the aperture function of the object, thus giving an out of focus ghost image, as shown in Figure 3c.