1. Introduction

A wide range of research has been conducted for reconstructing 3D objects in video manner based on the holography principle [1, 2].One of the important topics for this kind of 3D display is how to enlarge the image size $S$ for 3D objects, as well as how to enlarge the viewing-zone angle $\theta$ indicating the range where 3D objects can be seen. Naturally, it is desirable for both $S$ and $\theta$ to be large. If we denote the number of pixels of SLM here by $N_x$ horizontally and $N_y$ vertically, the pixel pitch by $p_x$ horizontally and $p_y$ vertically, and the wavelength of the reconstructed light by $\lambda$, then the image size $S_x$ and viewing-zone angle ${\theta }_x$ in the horizontal direction are given as follows.

Although it is desirable to increase the number of pixels $N_x$, the maximum number is limited for a single SLM in practice. Therefore, a space-division multiplexing and time-division multiplexing have been investigated. The former one is to increase the number of pixels using multiple SLMs. If we denote the number of pixels in the horizontal direction of each SLM by $N_x{}^{\prime }$ and denote the number of SLMs by $K_x$, then $N_x=N_x{}^{\prime }{K}_x$. The latter one is to change light path and update hologram data quickly to make it seem as if $N_x$ were increased. Some methods for the latter one have been studied such as the method to use shutters [3, 4] and the method to use a galvano mirror [5]. In addition, the Active Tiling^TM is a promising method that uses lens array and shutters to write hologram data on optically addressed SLM in sequence, which is also classable to time-division multiplexing [6]. Since it is known that the space-division and time-division multiplexing can be implemented together ingeniously [7], we deal with only a space-division multiplexing in this paper.

Various methods have been proposed for increasing the viewing-zone angle by a space-division multiplexing. These methods typically face with the problem that some parts of the viewing-zone angle are missing because it is physically impossible to locate the SLMs very closely together. Therefore, finding a means of eliminating this missing segment problem is a major issue. For example, a method of doubling the viewing-zone angle by using two SLMs and a beam splitter [8] and a method of tripling it by using three SLMs and beam splitters [7] have been proposed. A method of enlarging the viewing-zone angle by a factor of 12 in the horizontal direction has also been proposed by locating SLMs in an arc shape to project onto a vertical diffusing screen; however this method loses vertical parallax [9]. Yet another method provides two sets of multiple SLMs located in arc shapes, which are combined using a beam splitter [10]. All of the methods unfortunately cannot control the image size at the same time.

Some methods have also been proposed for increasing the image size by a space-division multiplexing. In these methods, finding a means of eliminating missing image segments is also a major issue just like it is for the methods of increasing the viewing-zone angle. For example, a method of tripling the image size has been proposed by using three SLMs and a large beam splitter and lenticular sheet [11]. In this method, the image size of each SLM and the gaps between SLMs must be the same. A concept using multiple modules that combine an enlargement optical system and a resolution conversion technique has also been proposed [12]. A problem with both of these proposals is that they only provide horizontal parallax.

In this paper, we propose a method for increasing the image size, and describe the device we developed on the method. The method locates an optical system containing a lens array and other components in front of multiple SLMs. It cannot only eliminate the missing image segments, but also control the balance between image size and viewing-zone angle by varying the focal length of lenses in the latter half within the optical system. In addition, it can exhibit parallax in both the horizontal and vertical directions unlike the methods mentioned earlier.

This paper is organized as follows. Chapter 2 describes the basic configuration of our method, Chapter 3 describes the particular specifications of the device and the experimental results, and Chapter 4 presents conclusions.

2. Configuration to eliminate the gaps between SLMs

Figure 1 shows the configuration of this device. This figure only illustrates an example of the horizontal direction for the case $K_x=3$. First, parallel light from a coherent light source is incident perpendicularly on SLM ${\text{C}}^0$ through beam splitter ${\text{B}}^0$. Consequently, this is so-called in-line holography. The hologram data is displayed on ${\text{C}}^0$. Note that the superscript at the top right of the letter represents the location within the array. Since the pixels are arranged in a grid pattern on a thin hologram, the light reflected on ${\text{C}}^0$ contains a primary beam, conjugate beam, carrier beam and high-order beams of each of them. Since the object beam is generated as a primary beam, the other beams are unnecessary light and must be eliminated.

 

Fig. 1 Configuration to eliminate the gaps between SLMs.

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Next, the light is incident on lens ${\text{L}}_{\text{0}}^{\text{0}}$, spatial filter ${\text{F}}^{\text{0}}$ and lens ${\text{L}}_{\text{1}}^{\text{0}}$. Spatial filter ${\text{F}}^{\text{0}}$ is located on the common focal plane of lenses and , and the focal length $f_0$ of and focal length $f_1$ of are related as follows, depending on the image size $S^{\prime }$ of each SLM and the gap $T^{\prime }$ between SLMs.

Here, let us simply denote the group of ${\text{C}}^0$, ${\text{B}}^0$, ${\text{L}}_{\text{0}}^{\text{0}}$, ${\text{F}}^{\text{0}}$ and ${\text{L}}_{\text{1}}^{\text{0}}$ by partial optical system ${\text{A}}^0$. By locating the partial optical system ${\text{A}}^0$ and ${\text{A}}^2$ adjacent to ${\text{A}}^1$, the images due to ${\text{C}}^0$, ${\text{C}}^1$, and ${\text{C}}^2$ will be tiled on plane ${\text{P}}_{\text{2}}$ without gaps. As a result, the missing image segments due to the gaps between the SLMs will disappear.

The group of lenses ${\text{L}}_{\text{0}}^{\text{0}}$, ${\text{L}}_{\text{0}}^{\text{1}}$, and ${\text{L}}_{\text{0}}^{\text{2}}$ forms a lens array ${\text{L}}_0$ in practice as well as the group of lenses ${\text{L}}_{\text{1}}^{\text{0}}$, ${\text{L}}_{\text{1}}^{\text{1}}$, and ${\text{L}}_{\text{1}}^{\text{2}}$ forms a lens array ${\text{L}}_1$. ${\text{F}}^{\text{0}}$ is a spatial filter for eliminating the conjugate beam, carrier beam, and high order beams, which are unnecessary light [13]. The group of spatial filters ${\text{F}}^{\text{0}}$, ${\text{F}}^{\text{1}}$, and ${\text{F}}^{\text{2}}$ forms a spatial filter array $\text{F}$ in practice.

Finally, the light is incident on lens $\text{L}{}_{\text{2}}$ and lens $\text{L}{}_{\text{3}}$. By setting the focal length $f_2$ of $\text{L}{}_{\text{2}}$ and focal length $f_3$ of $\text{L}{}_{\text{3}}$ as follows, the image that had been enlarged at ${\text{P}}_{\text{2}}$ will be returned to its original size at plane ${\text{P}}_{\text{3}}$.

The image on ${\text{C}}^0$ at ${\text{P}}_1$ is enlarged and appears at ${\text{P}}_{\text{2}}$without gaps. After that, the image at ${\text{P}}_{\text{2}}$ is shrunk and appears at ${\text{P}}_{\text{3}}$. The relation among viewing-zone angle ${\theta }_x$, total image size and total gaps, i.e. total missing segments, is shown in Table 1 .

Table 1. Relation among viewing-zone angle, total image size and total gaps

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If you set a fixed value for $N_x\lambda$ in Eq. (3) and you intend to vary the image size $S_x$ and viewing-zone angle ${\theta }_x$, you should set $f_2$ and $f_3$ to different values than in Eq. (6). Since that is a single optical axis, there also will be no missing image segments at ${\text{P}}_{\text{3}}$, just like at ${\text{P}}_{\text{2}}$. However, in this case, since the scaling factor in the optical axis direction differs from the scaling factor in other directions, you must provide hologram data that compensated for this discrepancy.

This configuration can exhibit parallax in both the horizontal and vertical directions. It can also control the balance between the image size $S_x$ and viewing-zone angle ${\theta }_x$ according to the values of $f_2$ and $f_3$. By varying the focal lengths of lenses, you can configure the ratio between the image size of each SLM and the gap between SLMs. The image size $S_x$ can be enlarged by increasing the number of SLMs $K_x$ and using a large-aperture lens for $\text{L}{}_{\text{2}}$. Unnecessary light can be eliminated by spatial filter ${\text{F}}^{\text{0}}$. Those are some advantages of this configuration.

The Active Tiling^TM also includes lens array [6]. In that method, the lens array is used to write hologram data on optically addressed SLM in sequence. Our method uses lens array to eliminate the missing image segments for space-division multiplexing, which is different from Active Tiling^TM.

3. The device we developed and experiments

We constructed a device with the parameters shown in Table 2 . This device, which uses an optical system and 9 SLMs, corresponds to a SLM with approximately 74,600,000 pixels. Figure 2 shows an exterior view of the constructed device. The arrows in the figure indicate the light path. We performed the following experiments using hologram data created by computer generated hologram (CGH) technology. Note that in this device, $T_x{}^{\prime }$ is three times the size of $S_x{}^{\prime }$ due to physical constraints of the SLMs. In other words, the missing area ($(S_x{}^{\prime }+T_x{}^{\prime })\times (S_y{}^{\prime }+T_y{}^{\prime })-S_x{}^{\prime }\times S_y{}^{\prime }$) is 15 times larger than display area ($S_x{}^{\prime }\times S_y{}^{\prime }$) as shown Fig. 3 . Therefore, observer cannot directly understand the reconstructed object, and an optical system like the one described here is required.

Table 2. Device parameters

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Fig. 2 Exterior view of the constructed device.

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Fig. 3 Display area and gap area.

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First, we confirmed the viewing-zone angle, the image size, and the fact that there were no missing segments. Figure 4 shows the experimental results. Figure 4(a) shows the design values of the displayed object on hologram plane. Figure 4(b), 4(c), and 4(d) show photographs that were taken of the reconstructed objects from locations 3.8 degrees to the right, directly in front, and 3.8 degrees to the left, respectively. It is apparent that the reconstructed objects can be observed in the designed viewing zone and missing segments that can be seen without the optical system are eliminated. In addition, Fig. 4(e) was photographed with real rulers located near the reconstructed objects. It is apparent that the reconstructed objects match the designed size, 9 times larger than the size of a SLM.

 

Fig. 4 Experimental results of simple graphic on hologram plane.

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The dark area can be ideally eliminated perfectly; however the dark area exists in Fig. 4(b) and 4(d) a little in practice. This is caused by aberration of the lens array ${\text{L}}_{\text{0}}^{}$. Since it can be eliminated on theory, we could eliminate it if we could remove the aberration. In addition, we could eliminate it if we use $f_1$ a little bigger than that of Eq. (4), which loses viewing-zone angle or image size a little but is practical.

Next, we confirmed that objects with different depths could be reconstructed. Figure 5 shows the experimental results. Figure 5(a) shows the design values of the displayed objects. Figure 5(b) and 5(c) show photographs that were taken of the reconstructed objects focused on the earth in the front and the moon in the back, respectively. When the objects can be reconstructed at different depths, the moon should be blurry in Fig. 5(b) and the earth should be blurry in Fig. 5(c). From the experimental results and the description above, it is apparent that objects with different depths can be reconstructed.

 

Fig. 5 Experimental results of 3D objects located at different depth.

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Finally, we confirmed that a video could be reconstructed. Figure 6 shows the experimental results. Figure 6(a) shows the design values of the displayed objects. The letter “C” located 30 mm behind the hologram plane gradually moves forward until it is 30 mm in front of the plane and then gradually moves back until it is 30 mm behind the plane. The video sequence shows the letters “N,” “I,” and “T” also sequentially moving forward and back in a same manner. This experiment confirmed that this video sequence can be played smoothly at a frame rate of 60 frames per second [fps] for approximately 15 seconds.

 

Fig. 6 Experimental results of video sequence (Media 1).

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Figure 6(b) shows a photograph that was taken of the reconstructed objects at the instant that the letter “C” is located 30 mm in front of the plane with the focus on the letters “N,” “I,” and “T,” while Fig. 6(c) shows a photograph that was taken with the focus on the letter “C.” It is apparent that depth can be reconstructed since letters at different depths are blurry. To see that this video can be played smoothly, see Media 1.

From the experiments described above, it is apparent that a video of 3D objects having an image size with a diagonal of 63 mm and a viewing-zone angle of 7.5 degrees can be played with frame rate of 60 fps using the proposed method.

4. Conclusions

A problem encountered when increasing the image size by using multiple SLMs is that some parts of 3D objects are missing due to the gap between SLMs. In this paper, we propose a method without missing image segments by locating an optical system containing lens arrays and other components in front of multiple SLMs. Advantages of our method are that parallax can be exhibited in both the horizontal and vertical directions, the balance between the image size and viewing-zone angle can be controlled, the device can be configured without regard to the ratio between the image size of each SLM and the gaps between SLMs, the image size can be increased by increasing the number of SLMs, and unnecessary light can be eliminated. We applied the method to 9 SLMs to realize an image size with a diagonal of 63 mm, a viewing-zone angle of 7.5 degrees, and a frame rate of 60 fps.