# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 4 Phase Aberration Compensation  
According to the basic principle of DHM, the first term in Equation (3) is zero when CCD locates at the image plane. Thus Equation (2) may be simplified as:  
\[O_{image}(x,y)=A_{0}\exp\biggl{[}-\frac{jk}{2f_{1}M_{0}}\cdot\biggl{(}x^{2}+y^ {2}\biggr{)}\biggr{]}O_{0}\biggl{(}\frac{x}{M_{0}},\frac{y}{M_{0}}\biggr{)}, \tag{14}\]  
where \(M_{0}\) is lateral magnification, i.e.,  
\[M_{0}=-\frac{d_{2}}{d_{1}} \tag{15}\]  
and the constant \(A_{0}\) is written as:  
\[A_{0}=\frac{\exp[jk(d_{1}+d_{2})]}{4\pi^{2}M_{0}}. \tag{16}\]  
Bringing Equations (5) and (14) into Equation (6), the filtered hologram is expressed as:  
\[ \begin{array}{c}I_{{Himage}^{\prime}}(x,y)=O_{image}(x,y)R^{*}(x,y)\\ =A_{0}\exp\Bigl{[}-\frac{jk}{2f_{1}M_{0}}(x^{2}+y^{2})\Bigr{]}O_{0}\biggl{(} \frac{x}{M_{0}},\frac{y}{M_{0}}\biggr{)}\cdot\exp[jk(x\cos\alpha+ycons\beta)]. \end{array}  \tag{17}\]  
It is seen that there always exists the two main phase distortions in DHM, which reflects the difference between the reconstructed phase and the measured phase. The first-order phase distortion is caused by the noncoaxis between the object and reference waves in off-axis configuration. It is given as:  
\[\xi_{1}(x,y)=\exp[jk(x\cos\alpha+y\cos\beta)]. \tag{18}\]  
The quadratic phase distortion is introduced by MO and its expression is shown as:  
\[\xi_{2}(x,y)=\exp\biggl{[}-\frac{jk}{2f_{1}M_{0}}\biggl{(}x^{2}+y^{2}\biggr{)} \biggr{]}. \tag{19}\]  
The two distortions result in the tilt and curvature that occurred in the reconstructed phase. Therefore, it is essential to apply a compensation approach to remove the phase aberrations. The phase aberration compensation approaches may be divided into two categories that are physical and numerical compensation methods. The physical compensation methods allow the optical configuration to realize the phase distortion elimination, such as the double exposure method [22; 63], the two same MOs inserted in the object and reference branches [25; 26; 29; 64], and telecentric arrangement [28; 30]. In 2003, [PERSON] et al. [22] proposed the double exposure method, which required recording two holograms. One was an object hologram with a sample, another was a reference hologram with a phase distortion in absence of a sample. The two reconstructed phases were subtracted to compensate for the distortions introduced by the optical components. It is ensured that the optical configuration is strictly the same in the two recording processes. Subsequently, the principle analysis and the experimental demonstration were further performed by [PERSON] et al. in 2009 [63]. The double exposure method by background subtraction is simple and practical. However, the application of the method is restricted for the sample with high-spatial-frequency content since it needs to select a blank region in absence of a sample as the reference. Aiming at the problem, [PERSON] et al. [65] presented an aberration compensation method using lateral shifting and spiral phase integration in 2017. Three holograms were captured by laterally shifting the samples in orthogonal directions. The subtraction of the original phase from the shifted phase images was completed to cancel the aberration out, as shown in Figure 4. Figure 4a shows the region selection results for the conventional background subtraction method and the shift differential method. In the conventional background subtraction method, a sample region in Figure 4b and a background region in Figure 4c are recorded. In the shift differential method, the sample region and its vertical and horizontal shifts are recorded. Figure 4b-d depicts the principle of the conventional background subtraction method, whereas Figure 4e-g provides the principle of shift differential method. It was shown that the presented method may be useful for measuring confluent samples, which was not available in the background subtraction method. But the requirement of the additional holograms restricted its application in dynamic measurement.