# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 4 Phase Aberration Compensation
The introduction of the MO enables the output to become the spherical wave in the pre-amplification configuration. When the interference happens between the spherical and plane waves, the hologram pattern is a set of circle fringe. This proved that quadratic phase distortion exists. In order to resolve the problem, [PERSON] et al. [64] inserted the same MO\({}_{2}\) in the reference branch as the MO\({}_{1}\) in the object branch in 2005, in which the MO\({}_{2}\)'s location was adjusted to offset the phase curvature in the object branch, as shown in Figure 5. The distance of \(L_{0}+L_{1}\) was equal to the distance of \(L_{0}+L_{2}\) as adjustable as possible to ensure the matching of the spherical waves in two branches. In addition, [PERSON] et al. [25; 26; 29] further carried out the research on the Michelson interference structure. They observed the spectrum distribution of a hologram to judge whether or not to remove the quadratic phase distortion. The method demands that only the distances between the MO and the CCD in the two branches are controlled precisely. The phase curvature can be eliminated. Figure 4: Principles of the shift differential method. (**a**) Region selection for conventional background subtraction method and shift differential method; (**b**) Phase image of a sample region; (**c**) Aberration image obtained from the background phase image; (**d**) Improved phase image, which is the subtraction of the aberration image from the phase image; (**e**) Horizontal shift differential phase image; (**f**) Vertical shift differential phase image; (**g**) Phase image retrieved from the spiral phase integration of differential phase images. (Reproduced with permission from [65], Copyright OSA publishing, 2017). In 2011, [PERSON] et al. [28] proposed a telecentric arrangement based on the DH pre-amplification configuration, as shown in Figure 6. The telecentric optical arrangement, composed of the MO and a collimated lens in the object branch, enabled the object wave to be plane from the spherical wave. The collimated lens with a larger focus length \(f_{2}\) and a smaller numerical aperture was located behind the MO. Its front focus plane coincided with the MO's rear focus plane, resulting in an afocal arrangement. The interference happened between two plane waves, thus compensating the quadratic phase distortion. When the object was located at the MO's front focus plane, the image would be produced in the rear focus plane of the collimated lens. The lateral magnification was estimated by the ratio of the lens's focal length \(f_{2}\) to the MO's focal length \(f_{1}\). The depth of the grooves of the zone plate was measured by the telecentric DHM. It was found that it agreed with the results obtained by profilometer. The method not only compensates the quadratic phase distortion, but enlarges the available region of spatial frequency in the off-axis configuration. In 2013, the same team [30] continued to analyze the telecentric arrangement in-depth. They demonstrated that the afocal arrangement possesses the shift-invariant imaging property, and the optical arrangement can remove the quadratic phase aberration. The numerical compensation methods mainly include a phase mask [22; 26; 60], the reference conjugated hologram [66; 67; 68], and the surface fitting method [23; 69; 70; 71; 72]. A phase correcting model at the hologram plane was established by [PERSON] et al. [22] in 2003, in which the correcting parameters were found by the flat area in proximity to the sample, but without the sample. The four correcting wave fronts with unitary amplitude but complex phases were given. The first three terms were considered to eliminate the circular fringes and slightly elliptical fringes, and the last term was used to remove the residual tilt. The reconstructed phase-contrast image of the sample and the reconstructed correction phase were obtained by propagation at distance \(z_{i}\), respectively. The two reconstructed phases were subtracted to obtain the sample information.