Holographic display is considered as the ultimate display as it can express a three-dimensional (3D) object by reproducing the wavefront of light. Since the advent of the hologram, efforts have been steadily made to replace the existing flat panel display [1–11]. However, the holographic displays have an inherent trade-off relationship between the viewing angle and the display size. Unlike the flat panel display using an incoherent light source, a hologram is generally reproduced by a coherent beam and a spatial light modulator (SLM) with pixelated structure. Here, the product of the diffraction angle and the area of the SLM is defined as the space-bandwidth product (SBP) that determines the overall performance of the holographic display system.

Typically, the SBP is equal to the number of pixels of the SLM. However, even with the 4K–8K resolution of the most advanced SLMs, there is a fundamental limitation of having a much smaller viewing angle and size than flat panel displays. Accordingly, with the innovation in the manufacturing industry to increase the number of SLM pixels, a lot of research has been actively done to overcome the bandwidth limitations of the holographic display.

The multiplexing technique in time or space can effectively extend the limited bandwidth of the SLM, so it has been widely used in holographic displays. However, the spatial multiplexing method using multiple SLMs is bulky and expensive [1]. In addition, for the case of the temporal multiplexing method, it requires relatively complex optics and consumes the system’s frame rate as the number of multiplexing [2,3]. Recently, several studies have shown that a scattering medium can be used to expand the viewing angle of the holographic display [4,5]. However, they require highly precise alignment between the scattering medium and the SLM.

In this Letter, we propose a novel method to increase the bandwidth of the holographic display using a multi-illumination strategy without temporal multiplexing. Typically, the effective range of the bandwidth is confined within the Nyquist frequency so as not to violate the sampling theorem. By illuminating multiple laser diodes (LDs) from different angles, the spatial frequency range of the hologram becomes wider with the extended energy distribution. However, the wavefront from each LD transfers identical information of the SLM that differs only in the carrier frequency, resulting in indistinguishable cross talk between the signals. Here, the main idea is to optically break the equivalence of the information by using a random binary mask (BM) at the frequency domain.

Figure 1 shows a schematic diagram of the proposed holographic display. It is based on a conventional holographic display with filtering optics. A multi-illumination is realized by adopting the LD array. Then, the BM in the spatial frequency domain acts as the filter and gives the degrees of freedom to modify the information from each of the LDs individually.

 

Fig. 1. Simplified concept diagram of the optical structure. Multiple collimated beams from laser diodes (LDs) are modulated in the phase-only SLM simultaneously. Then in the spatial frequency domain, the wavefronts are separately filtered with a BM to reduce their information similarity. After a 4$f$ system, the relayed SLM has extended bandwidth in proportion to the number of illuminations.

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A BM of arbitrary shape is considered in generating the computer graphic hologram (CGH). In this Letter, each direction (${i}, {j}$) of illumination is set to match with that of high diffraction orders as below:

The periodic pattern of the SLM generates high-order diffraction, which is the repeated pattern of the zero-order signal. Those repetition signals cause spatial aliasing artifacts that are considered noise and removed by filtering. However, in the proposed method, the intensity of the high diffraction order is further strengthened by the multi-illumination strategy to expand the bandwidth. In addition, each signal is filtered by the BM at different positions, which breaks the correlation of the signal. Finally, by making the duplicated information meaningful with the BM, it is possible to seamlessly extend the entire bandwidth of the holographic display over the number of illuminations.

Figure 2(a) illustrates our algorithm to synthesize the CGH. We optimize the phase of the SLM by using the stochastic gradient descent (SGD) approach. First, we consider the high-order diffraction [9], described as follows:

 

Fig. 2. (a) Schematic diagram of the proposed algorithm. The values in the brackets indicate the range of the spatial and frequency domains, where $N \times N$ is the SLM resolution. Note that the number of 2 in the third dimension indicates complex values. The wavefront $e^{i\phi }$ from the phase-only SLM is tiled in the frequency domain to account for the high-order diffraction. The spectrum $U$ is then multiplied by the binary mask $M$, transfer function $H$, and sinc amplitude $T_{ij}$. Each energy distribution of the ${(i,j)}\text {th}$ LD is calculated individually in the fourth dimension. After inverse Fourier transform, the reconstructed images ${|{u_{ij}}{|^2}}$ are summed on an intensity basis. Then, the loss is calculated to update the phase pattern taking the amplitude of the target and reconstructed result. (b) Illustration of the spatial frequency domain $(f_x, f_y)$. The multi-illumination angles are adjusted toward each central point of the high diffraction orders. Accordingly, the intensity is widely distributed over the extended bandwidth.

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The tiled spectrum $U$ is then multiplied by the binary mask $M\in \{ 1,0\}^{N_x \times N_y}$ and transfer function $H$ of the angular spectrum method after padding in the frequency domain [12]. With the propagation distance $z$, $H$ is defined as

We account for the carrier frequencies of the LDs and the fill factor of the SLM in the frequency domain by multiplying the sinc amplitude. When the SLM is illuminated by the $(i, j)\text {th}$ LD, the 2D sinc amplitude $T_{ij}$ is given by

From Eqs. (2)–(4), the complex amplitude $u_{ij}$ for the $(i, j)\text {th}$ LD at the distance $z$ is represented as

We verify the feasibility of the proposed method by measuring the reconstructed image quality using the peak signal-to-noise ratio (PSNR) and structural similarity (SSIM) index. To confirm the essential characteristic of the BM, we compare the proposed method against the single LD and multiple LDs cases without the mask. For the simulation, $3\times 3$ LDs with a wavelength of 635 nm and virtual SLM of $1000\times 1000$ resolution with 8 $\mu$m pixel pitch are used. The SLM fill factor $(a^2/d^2)$ is set to 0.92.

First, we investigated the properties of the BM pattern. For the simulation, the learning rate is set to 1.00, and the number of iterations is 1000. The phase value $\phi$ is initialized randomly in the range of $[-\pi,\pi )$ for optimization. Figure 3(a) shows that the image quality of the system is not significantly affected by the resolution of the BM. Here, very low-resolution BM may cause a noticeable discontinuity in the viewing angle. However, high-resolution BM requires precise alignment and decreases the optimization performance with a sparse spectrum. In addition, the BM with a lower density shows higher PSNR values as it increases the capacity to modify the information of each source; however, it penalizes the energy efficiency. Therefore, concerning the practical aspects, we chose the BM with $50\%$ density and $45 \times 45$ resolution, as marked with a red arrow.

 

Fig. 3. (a) Quality analysis of random BM. We used a “Dog” image at a propagation distance of 50 mm. (b) Quantitative evaluation according to the iteration number and the distance from the SLM. Box plot denotes the median, $25\text {th}$, and $75\text {th}$ percentiles. (c) Comparison of reconstructed images with and without the BM. The images with a dark background are presented to observe the noise by duplicated information better. In the case of not using the BM, “Dog” gets a lower PSNR as the background is darker than “Cat.” Source images by Agustsson et al. [13]. (d) Reconstructed images from the multi-illumination set of $(i, j)$. The information originating from the same SLM pattern is individually modulated and represents the target image propagating in different directions.

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Then, we investigated the reliability using 50 images in the DIV2K test dataset [13]. As shown in Fig. 3(b), our method achieves higher image quality than the cases where the BM is not applied. Even in the case without downsampling, it gives better performance. Although the performance can vary depending on the image used, our method not only yields a PSNR improvement but also is more robust near the SLM, where the information overlap is aggravated. Our approach can cover the wide depth range as the noise from the duplicated information is highly mitigated compared with the case without the BM. The simulation results verify that the proposed method could support high image quality with the multi-illumination strategy.

Figure 3(c) directly shows the validity of the random BM. In the case of not applying the BM, the duplicated images of the zero-order signal are placed in the shifted location according to each carrier frequency of LDs. In contrast, the proposed method effectively suppresses the noise originating from duplicated information. After each wavefront of the LD is blocked differently in the BM, the correlation sharing the single SLM is broken. As a result, the target image can be reproduced individually from the multiple illuminations. In Fig. 3(d), each ${(i, j)}\text {th}$ image represents the reconstructed result at the distance of 50 mm when the LDs are turned on one by one for the same phase pattern. The proposed method simultaneously reconstructs the image with multiple carrier frequencies, which in turn results in a wider viewing angle of the hologram with extended bandwidth.

Figure 4(a) shows the experiment setup. We used Thorlabs Exulus HD2 for the SLM and the CPS635F laser module with a wavelength of 635 nm. The $3\times 3$ LD array was assembled with a plastic housing made with a 3D printer. The BM was fabricated on glass (50 mm $\times$ 50 mm square, 2.3 mm thick) coated with patterned chromium. Three Nikon 105 mm camera lenses were used for the collimation and 4$f$ relay optics in the experiment. The results were captured using a CCD camera without lenses mounted on.

 

Fig. 4. (a) Photograph of the prototype. The manufactured LD array and random BM are shown at the bottom. (b) Captured results for a single-depth scene to demonstrate the feasibility of speckle noise reduction. The noise by duplicated information is well suppressed, as indicated in the dashed white circle. (Pieta: purchased 3D model from CGTrader.) (c) Enlarged distributions cropped from the results. We calculate the speckle contrast $C$ by dividing the standard deviation by the mean intensity.

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As provided in Fig. 4(b), the captured results show the novel feature of decreasing unwanted speckle noise with extended bandwidth. Overall, our method produces a face more smoothly compared with the standard case using a single LD source without the mask. Figure 4(c) presents the enlarged image cropped from the result. The speckle reduction effect is proportional to the overlap ratio of the regions illuminated by the LDs in the reconstruction plane. Since each central point of the high diffraction orders is placed at the edge of the image for a propagation distance of 50 mm, the speckle contrast reduces to approximately half without sacrificing the frame rate of the system as four independent speckle patterns are superimposed [2,3].

Furthermore, Fig. 5 demonstrates that the proposed method can expand the viewing angle and express 3D holograms. As shown in Fig. 5(a), the aliasing artifacts are effectively alleviated, and the viewing angle exceeds that of the single LD case, $\pm {2.3}^{\circ }$. We optimize the phase pattern with an amplitude loss function for two target planes to obtain the results of Fig. 5(b). The images are well reproduced experimentally at each depth.

 

Fig. 5. (a) Experimental results in changing the observation angles when the propagation distance is 0 mm. The results are captured using a CCD camera with a 100 mm lens (Visualization 1). (b) Experimental results of multi-depth scene. The distances are 50 mm and 53 mm, respectively.

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In summary, we propose a novel method to solve the limited bandwidth which has been a hurdle to the practical use of the holographic display. In this Letter, we introduce a new optical configuration using multi-illumination and a BM to enlarge the bandwidth of the holographic display. This approach effectively circumvents the trade-off between viewing angle and display size. Our experimental results demonstrate that the multi-illumination strategy achieved sufficient image quality with the extended bandwidth and effectively reduces the speckle noise. The maximum SBP of the method can be limited by the change in the characteristics of the SLM as the incident angle increases [14]. However, it was not a major concern in our experimental system, and more detailed analysis will be done in the future. The proposed method has practical significance as it is less sensitive to errors than the methods using a scattering device which requires a high-precision component and micro-scale alignment optics to improve the bandwidth. Even though it takes more time to synthesize the hologram with the expanded bandwidth, previous work has shown that training propagation models make real-time hologram synthesis possible [7]. Furthermore, our method can be integrated with previous works optimizing light sources to further reduce the speckle noise [8], and replace the random BM with a gray-scale mask to improve light throughput and image quality for future work [5].

We believe our concept of multi-illumination with the mask would be generally applicable to other systems such as near-eye or table-top holographic displays. We hope that our approach can contribute to the further development and realization of practical holographic displays.