## 3 Expressible Object Distance Range in a Hologram Depending on the Coherence
Radial blur is observed as shown in the result with a spatial filter size of 270 \(\upmu\)m of Figure 3. Therefore, the width of the voxel, \(w\), of the holographic display using the LED light source is the summation of the diffraction-limit width of the voxel, \(w_{diff}\), the increment of the voxel owing to the spatial coherence, \(w_{sc}\), and the increment of the voxel owing to the temporal coherence, \(w_{tc}\), as:  
\[w=w_{diff}+w_{sc}+w_{tc}=\frac{1.22\lambda_{c}|z_{obj}|}{h}+\frac{\lambda_{c} |z_{obj}|}{2\rho_{c}}+\frac{(|x_{obj}|+h)(\lambda_{max}-\lambda_{min})}{ \lambda_{c}}. \tag{6}\]  
Figure 5 shows the numerical reconstruction of voxels with different positions on object plane based on the spectral specifications of the LED light source. Here, 17 voxels with different positions in the object plane are selected and a Fresnel hologram is computed based on the central wavelength, \(\lambda_{c}\). Considering the spectral distribution of the LED light source, the increment of the voxel owing to the temporal coherence, \(w_{tc}\), is obtained as a weighted summation of the intensity profile of the numerical reconstruction by 1 nm wavelength intervals. Then, the voxel with both the spatial and the temporal coherence is computed as a convolution of \(w_{tc}\) and \(w_{sc}\). In this simulation, the objects distances \(z_{obj}\) are set to 50 mm and 100 mm, and the size of the spatial filter is set to 250 \(\upmu\)m and 1000 \(\upmu\)m. From Figure 5, it is obvious that the width of voxel increases accordingly as the size of the spatial filter increases. Also, the width of the voxel increases accordingly as the object distance increases. The blur of the reconstructed image from a low coherence is significant when the object plane is far from the hologram plane. For example, when the size of the spatial filter is 1000 \(\upmu\)m, the voxels located at 100 mm are too blurry. This fact gives some intuition that the holographic display has the expressible maximum object distance depending on the coherence of the LED light source. If the observer watches the holographic contents with a distance, \(z_{eye}\), from the hologram plane, the resolvable voxel width of the human eye, \(w_{eye}\), is defined as  
Figure 2:0  
where \(\theta_{eye}\) is the resolvable angle of the human eye, \(\frac{1}{60}\times\frac{\pi}{180}rad\). If the width of the reconstructed voxel, \(w\), is equal to or smaller than the resolvable voxel width, \(w_{eye}\), it can be considered that the holographic display is capable of expressing the voxel at a specific object distance, \(z_{obj}\). Consequently, the expressible object distance range of the holographic display is obtained using Equations (6) and (7), and it is represented as:Figure 2:1  
Figure 6 shows the change of the expressible object distance range depending on the spatial filter size in the holographic display using an LED light source. The relationship between the expressible object distance range and the spatial filter size is represented as shown in Figure 2:2 In this graph, \(z_{cye}\) and \(h\) are set to 1000 mm and 3.7 mm, respectively. In Figure 6, both the red and blue areas represent the expressible object distance ranges for the voxels with different transverse positions. When the transverse position of voxel \(x_{obj}\) is 0 mm, the expressible object distance depending on the spatial filter size is the red area. When \(x_{obj}\) is 1 mm, the expressible object distance is the blue area. Naturally, the expressible object distance range narrows as the spatial filter size increases. In addition, when the voxel is placed far from the optical axis, the expressible object distance range decreases because the radial blur is dominant in the off-axis. Figure 2:3 Numerical reconstruction of voxels with different positions on object plane and the spectral distribution of the LED light source. Figure 2:4 Expressible object distance range depending on the spatial filter size.