# Correlation Plenoptic Imaging: An Overview
## 2 Correlation Plenoptic Imaging with Chaotic Light--First Scheme
### Depth-of-Field Improvement
We show an image (**b**) refocused by using the correct value of \(z_{b}\), and two images (**c**,**d**) refocused at wrong distances \(z_{b}=60\,\text{mm}\) and \(z_{b}=90\,\text{mm}\), respectively. Reproduced with the permission from [67], copyright 2017 Society of Photo-Optical Instrumentation Engineers. Figure 5: Comparison between the incoherent images (**a**,**d**,**g**), the coherent images from CPI (**b**,**e**,**h**), and the refocused images from CPI (**c**,**f**,**i**), for three different single-slit masks of width: \(a=14\) μm = \(\Delta x^{f}\) (top panels), \(a=36\) μm \(\simeq 2.5\)\(\Delta x^{f}\) (central panels), and \(a=99\) μm \(\simeq 7.2\)\(\Delta x^{f}\) (bottom panel), where \(\Delta x^{f}\) is the diffraction-limited resolution in the focused image plane. The density plots report the correlation functions of Equations (13)–(15), normalized to their peak value. The solid (white) lines represent the size of the object, while the (white) dashed lines represent the tolerance on the blurring of the images, namely the resolution limit. The (black) dotted lines indicate the DOF limits, defined as values of the longitudinal distance at which the full width at half-maximum (FWHM) of the image increases with respect to the slit width \(a\) by the FWHM of the focused point-spread function. The setup employed for the simulation is the experimental one of Figure 10. The variations in the profile of large objects (**e**,**f**,**h**,**i**) are due to interference effects, related with the coherent nature of such images (from supplementary material of Ref. [42]). Reproduced with the permission from [42], copyright American Physical Society, 2017. Figure 6: Comparison between: **(a)** the incoherent image, (**b**) the coherent image from CPI, and **(c)** the refocused image from CPI, for a double-slit mask of width \(a=14\) μm = \(\Delta x^{f}\) and slit separation \(d=2a\). The density plots report the correlation functions of Equations (13)–(15), normalized to their value in \(x_{a}=0\), for any value of \(z_{b}-z_{a}\). The solid (white) lines represent the edges of the slits. The (black) dotted lines indicate the DOF, defined as the limits of \(z_{b}-z_{a}\) where the visibility drops below 10%. The images are from supplementary material, Ref. [42]. Reproduced with the permission from [42], copyright American Physical Society, 2017. Correlation Plenoptic Imaging offers the unique opportunity to refocus without sacrificing diffraction-limited image resolution, as defined by the numerical aperture of the imaging system. In Figure 7c, the dashed line represents the estimate, based on geometrical optics, for the maximum range of \"perfect\" refocusing in CPI [39], which is obtained from the condition  
\[\left|1-\frac{z_{a}}{z_{b}}\right|<\frac{\Delta x}{\Delta u}=\frac{dz_{a}/z_{b} }{\max[\lambda z_{b}/a,2\lambda/(M_{b}\,\mathrm{NA}_{b}),2\delta_{u}/M]}, \tag{16}\]  
with \(\Delta x\) the resolution on the spatial sensor D\({}_{a}\) and \(\Delta u\) the resolution on the source plane. On the right-hand side of Equation (16), we have considered \(d\) as the distance between object points that we want to resolve (e.g., the centers of a double slit), while \(a\) is the typical size of the smallest details of the object (e.g., the slit width). The resolution limit \(\Delta x=d\,z_{a}/z_{b}\) is defined by the geometrical projection of the image on the sensor plane. The resolution limit \(\Delta u\) is defined by the largest contribution among the ones determined by diffraction at the object (i.e., \(\lambda\,z_{b}/a\)), the numerical aperture of \(L_{b}\), or the pixel size \(\delta_{u}\); the last two contributions enter into play only for objects very close to the light source (i.e., for \(z_{b}=2\,a/(M_{b}\,\mathrm{NA}_{b})\) and \(z_{b}=2\,\delta_{u}\,a/(M\,\lambda)\), respectively).