## 3 Expressible Object Distance Range in a Hologram Depending on the Coherence  
In holographic displays, the voxel reconstructed by the displays is limited by the specifications of the optics. There are some significant factors that determine the voxel size. One is the diffraction limit of the holographic display. The other is the blur caused by the coherence of the light source. Figure 4a illustrates the width of the voxel in the holographic  
Figure 3: Reconstruction images of Fourier hologram depending on the various sizes of the spatial filter with 32.4, 75.6, 140.4, 205.2, 270.0, 507.6, 702.0, and 1090.8 \(\upmu\)m.  
Figure 2: Holographic display system with controllable spatial coherence using a digital micro-mirror device (DMD). (**a**) The scheme of the proposed system and (**b**) the experimental setup. display, considering the spatial coherence of the LED light source. First, the width of the voxel by diffraction limit, \(w_{diff}\), is determined as  
\[w_{diff}=\frac{1.22\lambda_{c}|z_{obj}|}{h} \tag{2}\]  
where \(h\) and \(z_{obj}\) denote the half width of the hologram and the object distance from the hologram plane, respectively. The voxel blur owing to the spatial coherence distance is depicted in Figure 4a. If the spatial coherence is represented by the coherence distance, \(\rho_{c}\), of Equation (1), the blur of the voxel owing to the coherence distance, \(w_{sc}\), is defined by  
\[w_{sc}=\frac{\lambda_{c}|z_{obj}|}{2\rho_{c}}, \tag{3}\]  
The temporal coherence of the light source also causes the blur of the voxel and it is closely related to the full width at half maximum (FWHM) of the LED light source in the intensity distribution according to the wavelengths. In general, the FWHM of the laser light source is only several nanometers and it is reasonable for it to be considered as a delta function in display applications. However, most color LED light sources with a central wavelength of \(\lambda_{c}\) have a FWHM of several tens of nanometers. Figure 4b depicts the blur of the voxel in the holographic display when the LED light source with some amount of the spectral bandwidth is used. The blur of the voxel according to the temporal coherence is easily estimated geometrically by considering the longest and shortest wavelengths in the FWHM of the LED light source. In general, the computer-generated hologram (CGH) is generated and displayed based on the central wavelength, \(\lambda_{c}\). When the voxel is  
Figure 4: The effect of coherence on the size of the voxel reconstructed in the digital holographic display. (**a**) The width of the voxel according to both the diffraction limit and the spatial coherence and (**b**) the width of the voxel according to the temporal coherence. reconstructed at the position, \(z_{obj}\), from the hologram plane, the position at which the voxel is reconstructed by the longest (\(\lambda_{max}\)) and shortest (\(\lambda_{min}\)) wavelengths is determined as:  
\[z_{obj,\,\lambda_{max}}=\frac{z_{obj}\lambda_{c}}{\lambda_{max}}, \tag{4a}\] \[z_{obj,\,\lambda_{min}}=\frac{z_{obj}\lambda_{c}}{\lambda_{min}}. \tag{4b}\]  
Equation (4a,b) also explain the reconstructed position of the voxel depending on the wavelength. In Figure 4b, from Equation (4a,b), the blur of the voxel by the temporal coherence \(w_{tc}\) is determined as:  
\[w_{tc}=\frac{(|x_{obj}|+h)(\lambda_{max}-\lambda_{min})}{\lambda_{c}}, \tag{5}\]  
where \(x_{obj}\) is the transverse position of the voxel along the \(x\)-axis. From Equation (5), it is confirmed that the blur by the temporal coherence increases as the voxel is farther away from the optical axis and as the FWHM of the spectral bandwidth increases. Also, this indicates that the temporal coherence of the light source affects the radial blur of the voxel in the object plane.