# Recent Progress on Aberration Compensation and Coherent Noise Suppression in Digital Holography
## 2 Basic Principle of DHM  
DHM allows for inserting a MO with different magnification in the object branch to enhance the lateral resolution, thus the interference happens between the spherical object wave and the plane reference wave in a pre-amplification configuration. The schematic of off-axis DHM pre-amplification configuration is shown in Figure 1. \(O(x,y)\) and \(R(x,y)\) represent the object wave and the reference wave, respectively. The object light propagates from the object input plane (\(x_{0}\),\(y_{0}\)) to the CCD output plane (\(x\),\(y\)), through the MO plane (\(x^{\prime}\),\(y^{\prime}\)).The quadratic phase factor introduced by MO is expressed as [54]:  
\[t(x^{\prime},y^{\prime})=\exp\biggl{[}-\frac{jk}{2f_{1}}\cdot\left(x^{\prime 2 }+y^{\prime 2}\right)\biggr{]} \tag{1}\]  
where \(k\) is the wave number \(2\pi/\lambda\), \(\lambda\) is the wavelength of the laser source, and \(f_{1}\) is the focal length of MO. The wave field propagation process is described based on the Fresnel diffraction. The object field \(O_{0}(x_{0}\),\(y_{0})\) goes through two Fresnel transformations, from input plane to MO over the distance \(d_{1}\), further to the output plane over the distance \(d_{2}\). The output field \(O(x,y)\) is derived as [55]:  
\[ \begin{array}{ll}O(x,y)=&-\frac{jk}{4\pi^{2}z^{\prime}}\cdot\exp[jk(d_{1}+d_ {2})]\cdot\exp\Bigl{[}\frac{jk}{2z_{1}}^{\prime}\cdot\left(x^{2}+y^{2}\right) \Bigr{]}\\ &\times\iint\limits_{\Sigma_{0}}dx_{0}dy_{0}O_{0}(x_{0},y_{0})\exp\Big{[} \frac{jk}{2z_{0}}^{\prime}\left({x_{0}}^{2}+{y_{0}}^{2}\right)-\frac{jk}{z^{ \prime}}\left(xx_{0}+yy_{0}\right)\Bigr{]},\end{array}  \tag{2}\]  
where  
\[\left\{ \begin{array}{ll}1/q_{M}=1/d_{1}+1/d_{2}-1/f_{1}\\ z^{\prime}=d_{1}d_{2}/q_{M}=d_{1}+d_{2}-d_{1}d_{2}/f_{1}\\ {z_{0}}^{\prime}={d_{1}}^{2}/(d_{1}-q_{M})\\ {z_{1}}^{\prime}={d_{2}}^{2}/(d_{2}-q_{M})\end{array} \right.. \tag{3}\]The interferogram produced by the output object wave and the reference wave is captured by CCD. Its intensity is expressed as:  
\[I_{H}(x,y)=\left|O(x,y)\right|^{2}+\left|R(x,y)\right|^{2}+O(x,y)R^{*}(x,y)+O^{*} (x,y)R(x,y), \tag{4}\]  
where the asterisk denotes the complex conjugate. The reference wave is written as:  
\[R(x,y)=\exp[-jk(x\cos\alpha+y\cos\beta)], \tag{5}\]  
where \(\alpha\) and \(\beta\) are the incident angle with respect to the \(x\) axis and \(y\) axis, respectively. In Equation (4), the first two terms are the intensity terms corresponding to the zero-order image. The third term and the fourth term are object and reference interference terms, respectively, including the amplitude and phase information. The zero-order image and conjugate image may disturb the object image, further influencing adversely on the imaging quality of the reconstructed image. Hence, the frequency filtering only retains the object image. The new hologram is written as:  
\[I_{H}{}^{\prime}(x,y)=O(x,y)R^{*}(x,y). \tag{6}\]  
A digital reference plane wave \(B\) is introduced to compensate the reference wave's conjugate in the new hologram. It requires that \(B\) should possess the identical tilt angle with the physical wave \(R\). Equation (6) is transformed as:  
\[U(x,y)=O(x,y)\cdot R^{*}(x,y)\cdot B(x,y). \tag{7}\]  
It is seen from Equation (7) that it only recovers the CCD output field, instead of the aerial image produced by the MO. The output field in the image plane (\(x_{i}\),\(y_{i}\)) should be obtained by Fresnel transform from the CCD plane to the image plane over the distance \(z_{i}\), then it is expressed as:  
\[U(x_{i},y_{i})=A\exp\biggl{[}\frac{jk}{2z_{i}}\left({x_{i}}^{2}+{y_{i}}^{2} \right)\biggr{]}\Im\biggl{\{}U(x,y)\exp\biggl{[}\frac{jk}{2z_{i}}\left(x^{2}+ y^{2}\right)\biggr{]}\biggr{\}}, \tag{8}\]  
where \(A\) is given as:  
\[A=\frac{\exp(jkz_{i})}{j\lambda z_{i}},\]  
and the corresponding Fourier transform frequency is written as:  
\[f_{x}=\frac{x_{i}}{\lambda z_{i}},f_{y}=\frac{y_{i}}{\lambda z_{i}}.\]  
Figure 1: Schematic of digital holography (DH) pre-amplification configuration.