Three-dimensional (3D) displays have the significant advantage of providing realistic images [1-4]. Among them, tabletop displays have the advantage of being able to produce a 3D image floating above a table [5-9]. These types of systems are based on multiview displays, and they are usually operated by temporal or spatial multiplexing methods to expand the viewing zone (VZ). The VZ is expanded by merging its sub-viewing zones (SVZ), from which an individual view is made. This implies that many SVZs are required to increase the viewing angle.

Both spatial multiplexing [10-14] and temporal multiplexing [15-17] are useful solutions to overcome the problem of insufficient amount of the information. In the spatial multiplexing method, the amount of information is increased using multiple spatial light modulators (SLMs). However, many optical components are required to implement spatial multiplexing, and the difficulty of optical alignment increases as the number of SVZs increases, while a tabletop display using the time-multiplexing method increases the amount of information for 3D content by adding only an optical scanner. Thus, 3D content is provided by using an SLM and an optical scanner at high speed. Although each method is a fundamental solution for expanding the VZ, there are practical restrictions on increasing the number of SVZs. For example, in a 3D display with only a small number of SVZs, the observer watches the discrete view over the full viewing angle. In other words, the motion parallax view looks discontinuous over the SVZs.

In multiview technology, there is no relation between the wavefronts of the adjacent views. In digital holography, however, there is a chance to overcome the limitation of multiview technology, since the wavefront is reconstructed from the hologram pattern on the SLM. Therefore, holography has been regarded as the ultimate technology [18-20], although it requires some conditions, such as a coherent light source, a spatial filter, and an SLM. The quality of the holographic image is determined by these conditions. Especially, the diffraction angle caused by the pixel pitch affects most display parameters, such as the viewing distance and viewing angle. Even though each wave is reconstructed individually in the temporal multiplexing method, the synthetic aperture can be constructed to solve the discontinuity of motion parallax.

In this paper, we propose a table-ornament 3D display using electronic holography with a synthesized VZ, which provides a viewing angle of 240 degrees. This table-ornament 3D display has a substantially smaller system volume than previously proposed tabletop displays. Our system is designed to decorate the table, has a small form factor, and provides a wide field of view for the people around the table. In the proposed system, 3f lenses are applied to holographically reconstruct an individual view. The combination of multiview technique and holography achieves an extremely continuous motion parallax. In addition, the time-multiplexing method reduces the burden of optical alignment. Figure 1 shows the schematic of the 3f parabolic mirrors used to form an imaginary object floating upon the system. Each ray from a point source is reflected by the upper and then the lower parabolic mirror sequentially. Then, the imaginary object appears to be located over the upper parabolic mirror. Thus, the optics provide floating 3D images to the observers, with good light efficiency due to the high reflectivity of the mirrors.

Figure 1. Schematic of the 3f parabolic imaging optics.

The proposed holographic table-ornament display is designed based on multiview and holographic technologies. Multiview is the primary technology used to expand the horizontal viewing angle, by connecting the VZ. For the table-ornament 3D display, 3f parabolic mirrors are used to generate a floating 3D image. In this section, the optical concepts and design of the table-ornament display are described.

Figure 2 shows the conceptual layout of the table-ornament display. The proposed system consists of four components: an SLM, Fourier-transform (FT) lens, optical scanner, and 3f optics. In Fig. 2, the 3f parabolic mirrors are replaced by 3f lenses to easily follow the trace of a ray. The wave from the SLM is projected in different directions by the optical scanner, and an individual view is formed in each direction. The FT lens is used to define the Fourier domain, in which the unwanted noise is filtered out. Then, the imaginary SLM is formed obliquely due to the optical scanner. This slanted imaginary SLM is imaged again by the 3f lenses, and the resultant image is formed on the optical axis. The VZ is placed on the circle located around the optical axis.

Figure 2. Conceptual layout of the table-ornament display for expanding the viewing zone. FT, Fourier-transform; SLM, spatial light modulator.

Figure 3 shows the layouts of two optical designs on the sagittal plane. This comparison let us know how much the imaginary SLM is tilted. Figure 3(a) shows the case when the imaginary SLM is parallel to the 3f lenses. Green arrows indicate the diffraction angle distributed around the surface normal vector of the SLM. The individual image is projected at the opposite side of the 3f lenses, and the SVZ area is defined by arranging an aperture stop for the SVZ connection. In this case the aperture stop is decentered from the optical axis, with the amount of decenter determined as follows:

Figure 3. Layouts of two optional designs, when (a) the SLM is placed in parallel with the 3f lenses, and (b) the SLM is oblique to the 3f lenses. SLM, spatial light modulator; SVZ, sub-viewing zones.

(1)$$x=\frac{f-u}{tan{\phi }_v}$$

Here f is the focal length of the single lens and u is the distance from the first lens. φ_v is the angle between the optical axis and viewing direction.

The projection of the aperture stop by the 3f lenses generates an SVZ’s viewing window (SVW). The distance of the SVW from the second 3f lens is given as

(2)$$v=f^2/u$$

where v is the distance between the second lens and the SVW. Consequently, the decenter of the SVW is given by

(3)$$x^{\prime }=x{M}_{total}=x{M}_1{M}_2=-x\left(f/u\right)$$

Here M_total is the total magnification of the 3f lens. M₁ and M₂ are the magnifications of the first and second lenses, respectively.

In Fig. 3(a), two problems are expected that are inappropriate for the table-ornament display. First, most ray bundles do not pass through the aperture stop, because the surface normal vectors of the imaginary SLM do not point toward the aperture stop. Second, the 3D image is perpendicular to the axis of rotation for scanning the SVZ, which causes no increase in the volume of the 3D image.

Figure 3(b) shows the other optional case, in which both the imaginary SLM and aperture stop are tilted with respect to the 3f lenses. Most ray bundles pass through the aperture stop, because both surface normal vectors of the imaginary SLM and the aperture stop have the same direction. However, the aperture-stop angle results in the rotation of the SVW. The lowest and highest positions of its boundary are drawn by blue and yellow dashed lines on the aperture stop. The images of these boundary points on the SVW are located along the s′₁ and s′₂ planes. The distances from the second lens are then derived using Eq. (2). The coordinates of the SVZ viewing are given by

(4)$$x_1^{\prime }=\frac{x-r_{stop}cos{\phi }_s}{u+r_{stop}sin{\phi }_s}f$$

(5)$$x_2^{\prime }=\frac{x+r_{stop}cos{\phi }_s}{u-r_{stop}sin{\phi }_s}f,$$

where r_stop and φ_s are the radius and angle of the aperture stop respectively. The angle of the SVW is given by

(6)$${\phi }_{SVW}={\tan }^{-1}\left(\frac{sin{\phi }_s}{ucos{\phi }_s-xsin{\phi }_s}\right).$$

The SVW angle suddenly increases, based on the angle of the aperture stop, when the aperture stop is decentered. It is important to consider the angles of both the imaginary SLM and aperture stop for light efficiency. Also, the mismatched viewing directions of the SVW need to be considered, which are related to the position from which the observer watches the content.

The imaginary SLM and aperture stop’s locations are determined on the sagittal plane for the table-ornament display, but the Wigner-distribution-function (WDF) analysis is based on the meridional plane. Although this analysis is helpful to determine the vertical viewing distance and the relationship between the aperture stop and SVW, it is necessary to analyze the sagittal plane for the expanded VZ. In the proposed system, the VZ is synthesized in the horizontal direction of the observer. In addition, the image quality from VZ synthesis is related to the operating speeds of the SLM, optical scanner, and optical structure. Even though the SVW is already determined by the aperture stop, the number of views is proportional to the operating speeds of the SLM and optical scanner. In other words, the crosstalk may be considerable when the SVW size is wide, for a large number of views. This phenomenon is simulated to analyze the relationship between the number of views and the SVW’s size for the proposed system.

Figure 4 shows the synthesized VZ of the proposed system. There are four eye positions. e₁ and e₃ are respectively the minimum and maximum viewing distances in the SVZ. e₂ and e₄ are the optimal viewing distances in the different SVZs. The viewing distance at which the observer watches an appropriate 3D image is the same as the decentered SVW. The SVW’s size is slightly different compared to the size in the meridional plane, because the slanted aperture stop’s length generates a different length in the SVW plane. However, the diameter of the aperture stop becomes its maximum, at which the horizontal SVW’s size in the sagittal plane is also maximum when the aperture stop has a circular shape with a small radius. The SVW’s size on the sagittal plane is expressed as

Figure 4. Synthesized viewing zone of the proposed system on the sagittal plane (e₁: minimum viewing distance, e₂: optimal viewing distance, and e₃: maximum viewing distance in a single SVZ, e₄: optimal viewing distance in another SVZ).

(7)$$w=r_{stop\left(f/u\right)}$$

The minimum interval angle between each SVW without crosstalk is given by

(8)$${\theta }_v=2{\tan }^{-1}\left(w/2x^{\prime }\right)$$

The minimum interval angle is the same as the diffraction angle on the 3D-image plane. The 3D image has large crosstalk when the interval angle is smaller than the minimum interval angle, but a black line appears in the 3D image when the interval angle is much larger than the minimum interval angle. The total viewing angle and viewing window of the expanded VZ are expressed as follows:

(9)$${\theta }_{total}=\left(N_{view}-1\right){\theta }_v$$

(10)$$w_{total}=x^{\prime }{\theta }_{total}$$

Here N_view is the number of single viewing windows.

The preceding analysis is simplistic, and insufficient to understand the VZ synthesis. On the other hand, the WDF analysis, which describes both spatial and spatial-frequency information, is very helpful for the purpose of visually comprehending this phenomenon [21]. The WDF is given by

(11)$$WDF\left(x,y;v_x,v_y\right)={\displaystyle \int {\displaystyle \underset{-\infty }{\overset{+\infty }{\int }}f\left(x+\frac{1}{2}x\text{'},y+\frac{1}{2}y\text{'}\right)\times f^{*}\left(x-\frac{1}{2}x\text{'},y-\frac{1}{2}y\text{'}\right)\times \exp \left[-i2\pi \left(v_{x}x\text{'}+v_{y}y\text{'}\right)\right]dx\text{'}dy\text{'}}}$$

Here, asterisk (*) represents the complex conjugate. The (x, y) pairs are the spatial coordinates and (ν_x, ν_y) are the spatial-frequency coordinates.

Figure 5 shows the results of the WDF analysis in the 3D-image plane and the synthesized VZ’s viewing-window plane. The WDF analysis based on our proposed system is shown in Fig. 3(b). The minimum interval angle is set at 0.6° and the diffraction angle is set at 0.3°, 0.6°, and 0.8°. Each gray region represents image information for a single viewing window, and white regions with colored borders represent the information observed by the observer’s eye. The green-bordered region indicates that the eye is located at the optimal viewing distance and in the SVW. The blue- and red-bordered white regions are near the optimal viewing distance, which is in the SVW, and a black-bordered white region is between two SVWs.

Figure 5. WDF analysis (a) in the 3D-image plane and (b) in the synthesized VZ’s viewing-window plane, when the diffraction angle of the 3D-image plane is larger than the minimum interval angle; (c) In the content plane and (d) in the synthesized VZ’s viewing window, when the diffraction angle of the 3D-image plane is the same as the minimum interval angle; And (e) in the 3D-image plane and (f) in the synthesized VZ’s viewing window, when the diffraction angle of the 3D-image plane is smaller than the minimum interval angle. WDF, Wigner-distribution-function; VZ, viewing zone.

In Fig. 5(a), the diffraction angle is greater than the minimum interval angle. Therefore, there is an overlapped area of information, which is represented by a dark gray color. This overlapped area of information is also related to the viewing window of the synthesized VZ. Therefore, all eyes receive the overlapped information, which causes crosstalk, as shown in Fig. 5(b). On the other hand, the case where the diffraction angle is smaller than the minimum interval angle is shown in Fig. 5(e). In this case, there is an empty window between SVWs. Thus, at the minimum viewing distance e₁, an individual image is received from the SVW. Meanwhile, at the optimal viewing distance e₂ and the maximum viewing distance e₃, the combination of the parts of a single image of each SVW is observed respectively, which results in the formation of a black line on the content. Even at the optimal viewing distance e₄, the observed image is sparser than the image from the SVW. However, in contrast to the previous two cases, crosstalk and black lines do not appear in Figs. 5(c) and 5(d). In other words, when the interval angle is almost the same as the minimum interval angle, 3D images are of the best quality without vignette.

Figure 6 shows a wide field-of-view table-ornament display using electronic holography. The optics are designed based on the analysis conducted in the meridional and sagittal planes. The diffracted light from the SLM propagates to the FT lens, and the light is filtered in the frequency domain, where the focal plane of the FT lens for the reconstruction hologram is located. The light is then folded by a folding mirror to a scanning mirror. The scanning mirror is designed to be free from the surface, which is called an extended polynomial surface in OpticStudio [22]. The surface sag of the extended polynomial is expressed as follows:

Figure 6. Layout of the wide-field-of-view table-ornament display using electronic holography. SLM, spatial light modulator; FT, Fourier-transform.

(12)$$z=\frac{c{r}^2}{1+\sqrt{1-\left(1+k\right)c^2{r}^2}}+{\displaystyle \sum _{i=1}^N{A}_i{E}_i}\left(l,m\right)$$

where c is the curvature. The E_i (l, m) is the i^th extended polynomial term in l and m, which are the coordinates at the aperture of the optical scanner. The A_i are the coefficients of the extended polynomial, and N is the number of polynomial terms in the series.

Finally, 3D content is located at the vertex of the upper parabola, and the light from the content is focused on the viewing window. The diameter and focal length of each parabolic mirror are approximately 212 mm and 75 mm respectively. The 3D-image size is approximately 39.3 mm, and the viewing direction is 44.2° from the optical axis.

The proposed table-ornament display using electronic holography is based on multiview and holographic technologies. Multiview is the primary technology used for expanding the horizontal viewing angle. The properties of the proposed system are determined by the operating speeds of the SLM, optical scanner, and optical structure. While the viewing window’s size is determined by the optical structure, the number of views is proportional to the operating speed of the SLM and the optical scanner.

The optical structure of the table-ornament display using electronic holography is designed by analyzing the meridional and sagittal planes. Although the scanner for the proposed system has a 360° scanning angle, the viewing angle of the proposed system is 240° because the other optical components are packed in the space between the two parabolic mirrors.

Figure 7(a) shows the structural schematic of the proposed system. Additional components used in the experiment include a collimator, digital micromirror device (DMD), folding mirror, and total-internal-reflection (TIR) prism. In the proposed system a DMD, which controls the on/off states of the pixels by tilting its micromirrors, is used for high-speed operation. A collimator is used to form the plane wave, and the TIR prism allows the light to be incident upon the surface of the DMD and blocks the light from the off pixels on the DMD.

Figure 7. The table-ornament display and its optical scanner. (a) Structure of the table-ornament display using electronic holography, (b) the scanning mirror, whose center of gravity deviates from the rotation axis, and (c) the scanning mirror with a modified center of gravity. DMD, digital micromirror device; TIP, total-internal-reflection.

Figures 7(b) and 7(c) show the structures of the scanning mirror. As mentioned above, the scanning mirror has an extended polynomial surface, because the plane of the required imaginary SLM is tilted. Therefore, this tilt of the scanner’s surface may result in mechanical vibration, due to the mismatch of the center of gravity with the rotation axis in Fig. 7(b). This mismatch is resolved by removing some volume from the backside of the scanning mirror, as shown in Fig. 7(c).

Figure 8(a) shows the lower part of the table-ornament display using electronic holography, in which two control devices are used to apply the time-multiplexing method. The proposed system uses as its DMD the V-7000 by ViALUX, with 1,024 × 760 pixels. The motor has a rotary encoder with 1024 pulses per cycle, and the trigger signal for the DMD is converted from each pulse using Arduino. Therefore, the number of single-viewing windows is 512. However, one-third of the viewing angle is not used in the proposed system, and approximately 341 SVZs are valid. The system’s volume is 210 × 210 × 203 mm³, and the diameter of the aperture at the upper parabolic mirror is 62 mm.

Figure 8. Photos of implemented system. (a) Lower part of the table-ornament display, which has the electronic control devices for the time-multiplexing method, and (b) the table-ornament display using electronic holography with 3f parabolic mirrors.

Figure 9 shows experimental results for the proposed system. In Fig. 9(a), the captured images represent the number of views in the target viewing angle. In the proposed system the SVW size is 3.7 mm, and the number of views in the target viewing angle is 341. But the number of views from the captured images ranges from 3 to 342. Experimentally 340 views are observed, and this number is almost same as the number of expected views, 341. The error is believed to be caused by the tolerance of cutting mirrors, and misalignment. There are curved-image distortions in views 173, 258, and 342. Although the design is based on the analysis at both meridional and sagittal planes, the image is slightly distorted. It results from the reduction of the number of optical components to make the system compact. Figure 9(b) shows a cone and a cylinder. In the target viewing angle, there is a disparity between the cone and the cylinder. Here, curved distortion also occurs, which is observed along the vertical straight line of the cylinder. The brightness of the image is not even, and the intensity at the center of the content is relatively high. This is mainly because the beam from the collimator has a Gaussian profile. The aperture size of the collimator is only slightly larger than the active area of the DMD, and the brightness of the image produced by the DMD also has a Gaussian profile. The viewpoints are provided by the time-multiplexing method. Likewise, it is possible to provide colorful content. For this purpose, an optical switch for an optical fiber is recommended. The optical switch selects the desired color before light of different colors from the lasers is delivered to the collimator.

Figure 9. Experimental results. (a) Captured images of the numbered individual views according to viewing angle from 0° to 240°, and (b) captured images of a cone and a cylinder according to viewing angle, from 0° to 240°.

In this paper, we have proposed a table-ornament electronic holography display using 3f parabolic mirrors. The proposed system has four main components: a DMD, FT lens, scanning mirror, and 3f parabolic mirrors. Multiview and digital holographic technologies are combined for continuous motion parallax. Consequently, the proposed system provides 340 views sequentially over a range of 240°. Moreover, the extended VZ has been analyzed with the help of the WDF to determine the appropriate number of views and the interval of the viewing window. The structure of the proposed system was analyzed in both the meridional and sagittal planes for optical design. In our system, the aperture stop is related to the viewing distance and the SVW size, and the optical scanner plays an important role in determining the optimal content. The SVW size in our system is 3.7 mm. The horizontal viewing window is increased by the time-multiplexing method, but the vertical viewing angle is narrow, which affects the viewing environment. The SVW size is determined by the Nyquist frequency; therefore, the pixel pitch of the SLM is related to the SVW size. It is possible to increase the vertical viewing angle by vertically diffusing or diffracting the information of the SLM in the horizontal-parallax-only display. We plan to increase the vertical viewing angle by using a diffraction grating in the plane of the imaginary SLM. In addition, the captured images are slightly distorted and the brightness of the image is uneven, due to nonuniform intensity from the collimator. In the future, we plan to enhance the image quality by a compensation algorithm for a computer-generated hologram.

The authors declare no conflicts of interest.

Data underlying the results presented in this paper are not publicly available at the time of publication, but may be obtained from the authors upon reasonable request.

This work was supported by a cross-ministry GigaKorea project (GK18D0100, Development of Telecommunications Terminal with Digital Holographic Table-top Display) grant funded by the Korean government (MSIT).

Korea government (MSIT) (GK18D0100).