## 2 Control of the Spatial Coherence in a Holographic Display System  
Digital holographic displays using an LED light source generally use a spatial filter to define the spatial coherence of the LED light source. Spatial coherence is closely related to the size of the light source, and a spatial filter restricts the size of the light source. Figure 1 describes the [PERSON]'s interference experiment using an LED light source with a central wavelength of \(\lambda_{c}\). Here, \(r_{s}\) denotes the diameter of the LED light source, and the double slit is positioned at a distance of \(R\) from the light source. The LED light source is considered as a collection of numerous point-light sources. Among them, three point-light sources, \(S_{1}\), \(S_{2}\), and \(S_{3}\), are chosen, where one is located at the center and the others are located at the ends of the LED light source. The spherical waves formed by each point light source pass through the double slit independently. Then, an interference pattern is formed by each spherical wave. If the interference patterns formed by \(S_{1}\) or \(S_{3}\) have an out-of-phase difference between them, the superposition of these two interference patterns is destructed. In this case, the distance between double slits becomes the coherence distance \(\rho_{c}\). Thus, the slit spacing needs to be smaller than the coherence distance, \(\rho_{c}\), to distinguish the interference from the LED light source. The relation between the size of the LED light source and the coherence distance is obtained by the geometric analysis as shown in Figure 1, and it is approximately determined as [20]:  
\[\rho_{c}\simeq\frac{R\lambda_{c}}{r_{s}}. \tag{1}\]  
In Equation (1), \(r_{s}\) denotes the diameter of the spatial filter to restrict the size of the LED light source. This implies that the spatial coherence increases as the diameter of the spatial filter decreases in the holographic display using an LED light source. However, when the coherence of the light source increases in holographic displays, speckle noise becomes severe and the quality of the reconstructed holographic content is degraded. Therefore, we devised the optical structure to control the spatial coherence in a holographic display system and it is possible to determine the optimal size of the spatial filter. Here, the spatial filter is defined by the on-state pixels of the DMD and the size of the spatial filter is controllable. Figure 2a shows the proposed holographic display system with controllable spatial coherence, which consists of an LED light source, imaging lenses, a DMD, a collimation lens, a beam splitter, an LCoS-phase-only spatial light modulator (SLM), and a Fourier lens. The LED light source has a central wavelength of 530 nm and maximum power of 20 mW. Using imaging lenses, the LED light source is imaged on the DMD. A LightCrafter with a DMD having the resolution of 608 \(\times\) 684 pixels, manufactured by Texas Instruments, is used as a dynamic spatial filter. Here, the lateral and diagonal pitches of the pixels are 7.638 and 10.8 \(\upmu\)m, respectively. The size of the spatial filter can be controlled by changing the on-state pixels of the DMD. The area of the on-state pixels is set to a circle and controllable as shown in Figure 2a. It is possible to increase the size of the spatial filter by 21.6 \(\upmu\)m, and it ranges from 32.4 to 1090.8 \(\upmu\)m. The LED light source filtered by the DMD is converted to the superposition of plane waves by the collimation lens and then it is modulated by the phase-only LCoS SLM. A collimation lens with a long focal length of 1 m is used to illuminate all active areas of the phase SLM. We used a phase-only LCoS SLM with a pixel pitch of 3.6 \(\upmu\)m and a resolution of 3840 \(\times\) 2160 pixels, manufactured by MAY display. Fourier holograms are used to confirm the feasibility of the proposed system. The effect of the spatial coherence is clearly observed since the Fourier hologram transforms the optical information of the hologram to the spatial frequency domain.