## 3 Making of the Primary Plane Hologgram
### Applying Digital Spatial Filter to the Calculated Primary Plane Hologram  
If the mm order's transferred holograms are simply overlapped partially, they will have much higher image intensity in the overlapped parts than the non-overlapped parts, which will cause split lines and reduce image quality. To improve this case, the intensity non-uniformity of the overlapped part should be counterbalanced to make it continue. The reconstructed image, which is the result of the first hologram calculated as a lensless Fourier hologram [9], should reduce image intensity near its edges. For a uniform intensity, the first hologram, which has been Fourier transformed, is multiplied with a spatial digital filter as shown in Figure 4a, then inverse Fourier transformed to obtain a filtered or intensity controlled hologram. The amplitude distribution of the filter is defined as:  
\[f(x,y)=f_{x}(x)f_{y}(y). \tag{1}\]  
Square of \(f_{x}(x)\) and square of \(f_{y}(y)\) are table shape function and written as:  
\[f_{x}(x)=\begin{cases}0&(x<-x_{1})\\ \sqrt{(x_{1}+x)/a}&(-x_{1}\leq x<-x_{1}+a)\\ 1&(-x_{1}+a\leq x<x_{1}-a)\\ \sqrt{(x_{1}-x)/a}&(x_{1}-a\leq x<x_{1})\\ 0&(x_{1}\leq x)\end{cases} \tag{2}\]  
where \(a\) is an overlapping width, and \(-x_{1}\) and \(x_{1}\) are left and right coordinates of the reconstructed image plane shown in Figure 3. The horizontal overlap ratio is defined as \(a/W\).  
\[f_{y}(y)=\begin{cases}f_{y}(|y|)&(0\leq y)\\ 0&(0<y<y_{1})\\ \sqrt{(y-y_{1})/b}&(y_{1}\leq y\leq y_{1}+b)\\ 1&(y_{1}+b<y<y_{2}-b)\\ \sqrt{(y_{2}-x)/b}&(y_{2}-b\leq y\leq y_{2})\\ 0&(y>y_{2})\end{cases} \tag{3}\]  
where \(b\) is an overlapping height, and \(y_{1}\) and \(y_{2}\) are bottom and top coordinates of the reconstructed image plane shown in Figure 3. The vertical overlap ratio is defined as \(b/H\). Figure 4a is a sample of digital spatial filter image in intensity (\(\propto f^{2}(x,y)\)) whose resolution is same as the first hologram (4096 \(\times\) 2160 pixel). Overlapping ratios defined below are 10% in horizontal and 10% in vertical. The white and gray parts are the window of reconstructed image plane. The black part is outside of the window with a value of zero. The middle white part is non-overlapped with a value of one. The gray part around the its edge is overlapped part with a value changes gradually from one to zero. Figure 4a shows the closeup of the top right filter area as marked with rectangle in Figure 4a. Based on Figure 4 and Equation (2), two different types of hologram tiling are shown in Figure 5a-c are normal edge in three kind joints and Figure 5d-f are filtered ones. Though in normal ones, well joint with edge as  
Figure 4: (a) Spatial filter to balance intensity on edges of the elemental volume hologram plane. (b) Closeup of the top right corner of the filter area as marked with rectangle in (a).  
shown in Figure 5b can solve the intensity problem and have good results in simulation, the quality of printed hologram also has problems in calculation of points data. The filtered ones, which is used to improve the quality, show three kind results in gap, aligned and overlapped. Figure 5d,f are insufficient overlap and excessive overlap. Similar with normal edge, the insufficient and excessive ones will cause the intensity distribution to be uneven--lower intensity with the black gaps between elemental holograms and higher intensity with overlaps. A comparison of these cases, shown in Figure 5e, is well aligned with the filtered edge. A filtered hologram can solve the problem of image intensity caused by misalignment and reduce the influence of the calculation problem.