Open Access
Add to BibSonomy
Abstract
We propose a compact type floating display system using a dihedral corner reflector array. Conventional floating displays using the dihedral corner reflector array usually have a folded configuration which makes the system bulky. The proposed technique achieves the compact in-line configuration using a pair of decentered lenses. The decentered lenses make the effective incident angle to the dihedral corner reflector array be tilted while maintaining the display panel and the dihedral corner reflector array in parallel. The ghost images are also refracted largely by the decentered lenses, being separated from the desired floating images. The proposed technique is verified by optical experiments.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Floating displays present images which appear to float in the air. Simple optical configurations using a display panel and a beam splitter have been frequently used. The image, however, is formed behind the beam splitter as a virtual one, preventing direct user interaction with the image [1]. Recently, floating displays using a dihedral corner reflector array (DCRA) have been reported [2–7]. The DCRA is a transmission-type device which performs retro-reflection along two transverse axes [8–11]. The light emanating from a point source in one side of the DCRA is converted into the light converging at its mirror-symmetric position in the other side, forming the real image. Since the image is real, users can reach and interact with the images unlike conventional beam splitter based systems.
One problem of the DCRA is that it forms not only the desired floating images but also the ghost images which are created by the non-desired reflections inside the DCRA structure. In order to separate the ghost images from the desired floating images, the DCRA based system usually has a folded configuration where the DCRA is tilted with respect to the display panel. However, this folded configuration makes the overall system bulky, limiting its applications [2–5]. A few works have been reported to avoid this problem. Y. Yoshimitsu et al. reported a radially arranged DCRA for a wide viewing angle [6]. The radial arrangement of the reflectors keeps the same incident angle over the DCRA plate, avoiding the ghost images. S. Choi et al. reported a ghost-free floating three-dimensional (3D) display using an offset lens with an integral floating display [7]. The off-axis integral imaging configuration increases the effective incident angle to the DCRA, contributing to the reduction of the overall system volume. These previous studies, however, still maintain the folded configuration that is hard to be implemented in a thin and compact form factor.
In this paper, we propose a novel compact in-line floating display system in which the display panel and the DCRA are aligned in parallel. A pair of decentered lenses makes the effective incident angle to the DCRA be tilted while maintaining the in-line compact configuration. The decentered lenses also refract the ghost images toward the outside of the viewing zone, increasing the effective floating image area. Therefore, the proposed system can present ghost free floating images around the system normal direction with a smaller system volume than the conventional configurations.
In the following sections, we first review the DCRA structure and the ghost image formation briefly. We then present the principle of the proposed system with experimental verifications.
2. Conventional method
2.1 DCRA structure and the image formation
The DCRA is an array of the corner reflectors. Figure 1 shows the structure of the DCRA used in the analysis and experiments in this paper. It consists of two layers each of which has a parallel plane strip mirror array. The mirror orientations of two layers are orthogonal, making ±45° with the positive x axis in the xy plane as shown in Fig. 1. The incident light rays undergo odd or even reflections in each layer, resulting in four different reflection cases.
Figure 2 shows the ray tracing results for a single point source. In the odd (lower layer) – odd (upper layer) reflection case, the rays are retro-reflected along the two mirror array axes, converging at the mirror-symmetric position of the point source as shown in Fig. 2(a). This reflection case contributes to the formation of the desired floating images. In the odd-even or even-odd reflection cases, however, the rays undergo the retro-reflection only along one of the lower- or upper-layer mirror axis. Therefore, the reflected rays converge toward the mirror-symmetric position only in that mirror axis while diverging from the original point source position in the other mirror axis as shown in Figs. 2(b) and 2(c). These two reflection cases create two corresponding ghost images with strong astigmatism aberration. Finally, in the even-even reflection case, the rays keep their original directions without retro-reflection. They diverge from the original point source position as shown in Fig. 2(d) and contribute to the direct observation of the display panel through the DCRA, which will be called a DC image in this paper.
2.2 Folded system configuration
Conventional floating displays using the DCRA usually have a folded configuration to separate the ghost and DC images from the floating image. In a simple in-line configuration shown in Fig. 3(a), the floating image is overlapped with the DC image from the even-even reflection. The rays from two ghost images also come into the eye, making them visible as shown in Fig. 3(b). This overlapping of the DC and the ghost images onto the desired floating image prevents the use of the in-line configuration. In the folded configuration shown in Figs. 3(c) and 3(d), the DC and the ghost images are well separated in the observation direction. The folded configuration, however, requires large system volume at a given image size, making it hard to be implemented in a compact form factor.
Fig. 3. Floating images according to DCRA configurations. (a) Non-folded configuration, (b) experimental result (non-folded configuration), (c) folded configuration, (d) experimental result (folded configuration).
3. Proposed method
3.1 System configuration
The proposed system has a non-folded in-line configuration with reduced system volume and well-separated ghost and DC images. As shown in Fig. 4, the proposed system uses a pair of symmetric decentered lenses located at the input and output sides of the DCRA. The decentered lens pair makes the beam angles toward and from the DCRA be tilted while the floating images are still formed normally around the system axis. The tilted beam angle around the DCRA contributes to the separation of the ghost and DC images from the desired floating images.
Figure 5 shows geometry to calculate the position of the floating, DC, and ghost images. Suppose that two convex lenses of the same focal length f are positioned around the DCRA with an axial separation a from the DCRA and a decenter c from the system axis, locating their centers at (±a, -c) in the (z, y) coordinates system defined in Fig. 5. Also suppose that a display panel is located at a distance b (<f) from the 1st decentered lens. Light rays emanating from a point source at (z1, y1)=(-(a + b), y1) in the display panel are refracted by the 1st decentered lens to form the virtual image at (z2, y2) where
The refracted rays then pass through the DCRA, experiencing 4 reflection cases.
In case of the odd-odd reflection, the rays exiting the DCRA converge toward the mirror-symmetric position (z3, y3)=(-z2, y2). The 2nd decentered lens then refracts these rays to form a real image at (zoo, yoo). From the lens equation and Eqs. (1)–(2), the odd-odd reflection real image position (zoo, yoo) is easily obtained by
Therefore, in the proposed configuration, the odd-odd reflection creates the floating image of the display panel at its mirror-symmetric position with respect to the DCRA, regardless of the decentered lens pair.
In the even-even reflection case, the rays exiting the DCRA keep their original directions, diverging from the virtual image point at (z2, y2). These rays are finally refracted by the 2nd decentered lens to form a real or virtual image. From the lens equation and Eqs. (1)–(2), the position of the real or the virtual image (zee,yee) is calculated to be
In the even-odd or odd-even reflection case, the rays converge toward (zoo, yoo) in one mirror array axis while diverging from or converging toward (zee, yee) in the other mirror array axis. Note that the mirror array axes are 45° rotated in the DCRA plane, i.e. z=0 plane, as shown in Fig. 1.
3.2 Single point source analysis
Figure 6 shows the ray tracing results for a single point source at the center of the display, i.e. located at (x1,y1,z1)=(0,0,-(a + b)). In the simulations shown in Fig. 6, the aberration-free ideal lens model is used for the decentered lens pair. In each of Figs. 6(a)–6(d), the ray tracing in z-y plane of the odd-odd and even-even reflection rays is shown in the left part, and the footprint of all odd-odd, even-even, odd-even, and even-odd reflection rays in the floating image plane, i.e. z = zoo, is shown in the right. The focal length of the decentered lenses is assumed to be f=177.8 mm in Figs. 6(a)–6(c) and it is varied in Fig. 6(d). Figure 6(a) shows the effect of the lens decenter c. When c=0 (no decenter), the floating image point formed by the odd-odd reflection is overlapped by the DC and the ghost images, failing to be separated. As the decenter c increases, the DC and two ghost images move away from the floating image point which remains at the same position regardless of c. Therefore, large decenter c is desirable to have sufficient separation. Figure 6(b) shows the effect of the display distance b from the 1st decenter lens. The angular separation of the even-even reflection DC image from the desired odd-odd floating image is fixed regardless of the display distance b as revealed in the left part of Fig. 6(b). The even-odd and odd-even ghost images are overlapped with the floating image when b=90mm≈f/2 and moves away from the floating image as b increases to f. This suggests that the display distance b should be sufficiently larger than the half of the focal length of the decentered lens. Figure 6(c) shows the effect of the gap a between the decentered lens and the DCRA. It is confirmed from Fig. 6(c) that a does not have much impact on the ghost and DC images separation, and thus a is set to be 0 in the experimental verification to minimize the system volume. Finally, Fig. 6(d) shows the effect of the focal length f. As the focal length f decreases, the separation from the ghost and DC images increases, which is preferable. It should be noted, however, that the small focal length also limits the maximum floating image distance b, which will be revealed in the next section. In our experiments, f=177.8 mm focal length was empirically selected and used with an acceptable compromise.
Fig. 6. Single point source ray tracing simulation of the proposed system configuration with different (a) decenter, (b) display distance, (c) gap between the lens and the DCRA, and (d) focal length of the lenses.
3.3 Extended display area analysis
Figure 7 illustrates the viewing zone of the proposed system. The viewing zone is defined by the angular range where all parts of the floating images are observed without intrusion of the ghost or DC images. In order to examine the observation angle dependent intrusion of the ghost and DC images onto the floating image, the ray tracing is performed in reverse direction for a set of parallel output rays. In the ray tracing simulations presented hereafter, instead of the aberration-free ideal lens model, a thin plano-convex lens model is used and the refraction at each side of the lens is considered for accurate results. Note that the actual experimental setup uses plano-convex Fresnel lenses which can be modeled as the thin plano-convex lens. The orientation of the plano-convex lenses in the simulation also follows the actual experimental setup where convex sides of the lenses face the input and output sides of the system respectively as shown in Figs. 4 and 5. The display is now considered as an extended source with 50mm×50 mm size, i.e. |x|≤25 mm and |y|≤25 mm at z=-(a + b). The gap a between the decentered lens and the DCRA is set to be zero, i.e. a=0, the amount of the decenter is c=60 mm and the focal length of the decentered lens is f=177.8 mm.
Fig. 7. (a) 2D illustration of the viewing zone. (b) Illustration of the reverse ray tracing for parallel output rays and the arrangement of the simulation results in Fig. 8.
For the simulation, parallel light rays passing through the floating display area, i.e. |x|≤25 mm, |y|≤25 mm at z = zoo=a + b at a specific angle θx and θy are traced back to the display plane in different DCRA reflection cases to see if they meet the valid display area, i.e. |x|≤25 mm and |y|≤25 mm at z=-(a + b) as illustrated in Figs. 7(b) and 8(a). This process is repeated for different observation angles θx and θy and the results are tiled together in Figs. 8(b)–8(g) to easily identify the angular range where the 50mm×50 mm display area is observed without intrusion of the ghost images.
Fig. 8. (a) Example of the simulated view at θx=0°, θy=-23° observation direction (yellow and orange colors represent ghost images). (b)-(g) are tiled collection of the simulated views at different observing directions when (b) b=90 mm, (c) b=135 mm, (d) b=170 mm, (e) b=200 mm, (f) b=250 mm, and (g) b=300 mm.
Figures 8(b)–8(g) show the results for different display panel distances b. When b=90 mm shown in Fig. 8(b), the two ghost images represented by orange and yellow color intrude the floating images represented by blue color when θy is negative. This limits the viewing angle of the floating image only to positive θy range, which is restrictive. When b=170 mm shown in Fig. 8(d), the ghost image intrusion into the floating images in the negative θy angles is not observed. At largely negative θy, however, the rays meet the decentered lenses at high y=(a + b)tan(-θy), i.e. large height from the lens center at y=-c, being lost by total internal reflection inside the lens which is modelled as the plano-convex lens in our simulation. Figure 8(d) also reveals that the intrusion of the ghost images is observed at large positive θy angles. Therefore b=170 mm is not found to be optimum in our simulation. Additionally, Figs. 8(e)–8(g) show that the intrusion of the ghost image becomes more severe as the b increases over the focal length f. Therefore, it is advantageous to keep the distance b less than the focal length f. The best result is found when b=135mm≈3f/4. As shown in Fig. 8(c), the 50mm×50 mm display area is well observed without the ghost images or light loss for most angular ranges in |θx|,|θy|≤23°. This result is verified by the experimental results in the following section.
4. Experimental results
The proposed configuration is verified experimentally. A glass ASKA3D-plate of 200 × 200 × 5.6 mm size is used as the DCRA [12]. A display of 1440 × 2560 resolution and 47.6µm pixel pitch is located at different distances b from the 1st decentered lens, i.e. b=90 mm, 135 mm, and 170 mm. In the entire 1440 × 2560 resolution display screen, central 1050 × 1050 pixel area which corresponds to 50mm×50 mm is used to display images. For the 1st and 2nd decentered lenses, instead of crafting lenses to obtain the decentered lens segments, we used the whole lenses and masked unnecessary parts leaving the effective decentered lens segments. The lenses are Fresnel lenses with 279.4 × 279.4 mm size and 177.8 mm focal length. They are attached to the input and output sides of the DCRA without a gap, i.e. a=0 and shifted downward to give 60 mm decenter, i.e. c=60 mm. The plano-sides of the Fresnel lenses are attached to the DCRA, making the convex sides of the lenses face the input and output side of the system, respectively. Figure 9 shows the implemented experimental setup and Table 1 lists their specifications.
Table 1. Experimental setup specification
Figure 10 shows examples of the experimental results which are captured at a normal direction, i.e. θx=θy=0°. When the mask is not used as shown in Fig. 10(a), not only the desired floating image but also two ghost images and the DC image are observed from the unnecessary parts of the whole lenses. By adding the mask which effectively defines the decentered lens segment, these unwanted images are blocked, leaving only the desired floating image in the normal observation direction as shown in Fig. 10(b).
Fig. 10. Examples of the floating image by the proposed method. The pictures were captured (a) without and (b) with the mask defining effective decentered lens segments. The display distance b is 135 mm and the picture were captured at normal observation direction.
Figure 11 shows the floating images with different display distances b and different decenters c. The experimental results shown in Fig. 11(a) were captured at normal observation direction, i.e. θx=θy=0°, and the simulation results shown in Fig. 11(b) were also obtained by the same condition, i.e. by tracing back the θx=θy=0°normal parallel light rays from the floating image area to the display panel as in the previous sub-section 3.3. When capturing the pictures in Fig. 11, the mask was intentionally removed to reveal the positional relations between the ghost images and floating images more clearly. The experimental results in Fig. 11(a) show that the floating images are severely overlapped by the ghost images at a small decenter c=30 mm. This overlap is, however, removed by increasing the decenter c, and the floating images are well separated from the ghost images at a large decenter c=60 mm in all display distances b as shown in Fig. 11(a). These experimental results also agree well with the simulations shown in Fig. 11(b) where the floating image is represented by the blue color and two ghost images are represented in orange and yellow colors. Therefore, the formation of the floating images without the ghost image intrusion by the proposed in-line configuration is confirmed.
Fig. 11. Floating images with different display distances b and the decenters c. (a) Experimental results captured at the normal direction θx=θy=0°, and (b) simulation results using θx=θy=0° normal parallel light rays passing through the floating image area masked by red-dotted rectangle in (a).
Figure 12 shows the formation of the desired floating images at their designated distances by the proposed in-line configuration. In Fig. 12(a), the display is located at 90 mm distance from the 1st decentered lens, i.e. b=90 mm, such that the floating image is formed at 90 mm distance from the 2nd decentered lens. The three pictures in Fig. 12(a) were captured with the camera focus located at 90 mm, 135 mm, and 170 mm from the 2nd decentered lens. It can be observed in Fig. 12(a) that the observed floating image is focused when the camera focus is at 90 mm, confirming the floating image is formed at that distance as expected. Figures 12(b) and 12(c) are the results when the display is located at b=135 mm, and b=170 mm, respectively. They also show that the floating image is formed at the corresponding distances, i.e. b=135 mm and b=170 mm, respectively. Therefore, the formation of the floating images by the proposed in-line configuration is confirmed experimentally.
Fig. 12. Floating images captured with different camera focus. The display distance is (a) b=90 mm, (b) b=135 mm, and (c) b=170 mm. In all cases, the amount of the decenter is c=60 mm and the pictures were captured in the normal direction θx=θy=0°.
Fig. 13. Floating images captured at 9 observation directions. The display distances are (a) b=90 mm, (b) b=135 mm, (c) b=170 mm, and (d) b=300 mm. In (a)-(c), 9 observation directions correspond to (θx= -23°,0°,23°, and θy= -10°,0°,10°). In (d), 9 observation directions are limited to (θx= -15°,0°,15°, and θy= -4°,0°,4°) due to the DCRA and lens size limitations.
Figure 13 shows the floating images observed from different directions for four display distance cases, i.e. b=90 mm, 135 mm, 170 mm, and 300 mm. In the normal observation direction θx=θy=0°, all four b cases show clear separation of the floating images from the ghosts. At large observation directions |θx|=23° and/or |θy|=10°, however, the floating images of b=90 mm and 170 mm cases are intruded by the ghost images, limiting their viewing angle. The large floating image distance case b=300 mm over the focal length f gives worse results showing large ghost image intrusion even at small |θx|=15°, |θy|=4° viewing angle. Therefore b=135mm≈3f/4 is found to give the largest viewing angle over ±23° horizontally and ±10° vertically in our experiment and this also agrees well with the simulation results in Fig. 8 and the right part of Fig. 13.
Fig. 14. Integral imaging experimental results. (a) Elemental images, (b) system configuration, (c) experimental results (distance b: 135 mm, depth d: -40 mm, 40 mm) observed from 9 directions (θx= -10°,0°,10°, and θy= -10°,0°,10°). The camera was positioned at 300 mm distance from the floating image and translated over +-50 mm range in transverse directions. The F/# of the camera was 10 which is similar to that of the eye. See Visualization 1 for a movie.
Finally, Fig. 14 and Visualization 1 show the extension of the proposed method to the floating 3D images. For the 3D display, integral imaging 3D display system is used instead of the 2D display panel [13–14]. The integral imaging system is implemented by adding a lens array of 1 mm lens pitch and 3.3 mm focal length onto the display panel. Two images ‘apple’ and ‘banana’ with depths d=+40 mm and d=-40 mm are displayed around the display panel which is at the distance b=135 mm from the 1st decentered lens. These two images are relayed by the proposed in-line configuration to form the 3D floating images at 135 mm ± 40 mm distances from the 2nd decentered lens. Figure 14 and Visualization 1 show that the floating 3D images are formed successfully, demonstrating clear motion parallax between ‘apple’ and ‘banana’ images.
5. Analysis
The proposed in-line floating display system has been verified successfully by the numerical ray tracing simulations and optical experiments implemented with the design parameters a=0 mm, b=135 mm, c=60 mm, and f=177.8 mm. For an analysis on the full design space of these parameters, in this section we present analytic formulas of the ray trajectory in the system. Note that in order to find simple and intuitive analytic formula, the decentered lens pair is assumed to be ideal lenses without any aberrations. This assumption is not valid in the actual system which has large decenter c=60 mm and large ray angles up to +-23° at the same time, being well outside of the paraxial regime. However, unlike the numerical ray tracing considering the refraction at each lens surface of the plano-convex lens, the analytic formula indicates the dependencies between the parameters and the system performance explicitly, helping the design of the system.
Figure 15 shows the geometry to find the ray trajectory. Suppose a ray BA which passes through a point B(xB,yB,a)=B(xB,yB,0) in the second decentered lens and the point A(xo,yo,a + b)=A(xo,yo,b) in the floating image. For this ray, we find the corresponding 4 points Eoo, Eee, Eeo, and Eoe in the display panel plane z=-(a + b)=-b by considering 4 different reflection cases of the DCRA. The gap a between the decentered lens and the DCRA is assumed to be zero in this section for simplicity. Considering the second decentered lens is the aberration-free ideal lens with its center located at (0,-c,a)=(0,-c,0), the direction of the ray CB corresponding to the ray BA is represented by a vector
Equations (12)–(15) can be used to find the condition for the design. For a specific viewing direction θx and θy, the points Eee, Eeo, and Eoe corresponding to a point in the floating image (xo, yo, b) are found by letting (xB, yB)=(xo-btanθx, yo-btanθy) in Eqs. (13)–(15) as
For a given parameter set b, c, f, and the floating image size |xo|<Lx/2, |yo|<Ly/2, the available viewing angle wherein the ghost images do not intrude the floating image is found from the conditions
Figure 16 shows a few examples of the (xB, yB) range calculated using Eqs. (16)–(18) which satisfies the condition of Eq. (19) for a given viewing angle range and the panel size. The (xB, yB) range in Fig. 16 can be regarded as the effective area of the DCRA and the decentered lens pair, defining the maximum numerical aperture of the system.
Equations (16)–(18) reveal that b/f close to 1 and large bc/f are advantageous to satisfy Eq. (19). Therefore, large decenter c, small focal length f, and the panel distance b close to f are generally preferable. However, it should be noted that as the panel distance b needs to be smaller than the focal length f as revealed in Figs. 8 and 13, the small focal length f leads to small floating distance, which degrades the floating effect of the system.
More importantly, Eqs. (7)–(18) are derived from the aberration-free ideal lens assumption which is not highly valid in the proposed non-paraxial decentered system. Figure 17 shows ray tracing comparison between the ideal lens model, bi-convex lens model, and the plano-convex lens model. Note that the actual lens used in the experiment has plano-convex type Fresnel surface and the numerical ray tracing simulations in sections 3 and 4 use the corresponding plano-convex lens model. It is obvious from Fig. 17 that the simple ideal lens model used in this section for the derivation of the analytic formulas shows large deviation from the actual model. Figure 18 shows the observation-direction-dependent ghost image intrusion analysis for different decenters using the ideal lens model and the plano-convex lens model. It is found from Fig. 18 that as the decenter c increases, the ideal lens model gives larger intrusion than the plano-convex lens model. Considering that the ray tracing using the plano-convex lens model agrees well with the experimental results as revealed in section 4, the ideal lens model used in this section gives stricter condition than the actual experiment. Therefore the condition given in Eq. (19) calculated from Eqs. (7)–(18) in this section needs to be used only as a rough estimation. Nevertheless, the analytic formulas of Eqs. (16)–(19) reveal the relationship between the parameters, guiding the design of the system.
Fig. 17. Point source array ray tracing simulation using different lens models. (a) Side views of three panel distance cases, i.e. b=90 mm, 135 mm, and 170 mm. The simulated footprints obtained using (b) ideal aberration-free lens model, (b) bi-convex lens model, and (c) plano-convex lens model with the same face orientations as the experimental setup.
Fig. 18. Comparison of simulated floating images at 9 observation directions with (θx= -23°,0°,23°, and θy= -10°,0°,10°) obtained using the plano-convex lens model (left) and the ideal lens model (right). The yellow and the orange parts show the intrusion of the ghost images. The display distance is b=135 mm, lens focal length is f=177.8 mm. The amount of decenter is (a) c=30 mm, (b) c=40 mm, (c) c=50 mm, and (d) c=60 mm.
6. Conclusion
A novel in-line floating display system using a DCRA and a decentered lens pair is proposed. The decentered lens pair makes the incident angle to the DCRA to be slanted so that the ghost images are shifted outside of the viewing area leaving the desired floating image unchanged. In-line configuration enables more compact system configuration than the conventional folded configuration. The proposed system is verified by ray tracing simulations and optical experiments, demonstrating formation of the floating images with ±23°(horizontal)×±10°(vertical) viewing angle around the normal system axis. Combination with the integral imaging system is also experimentally demonstrated, showing floating 3D images with clear motion parallax.
Funding
National Research Foundation of Korea (2017R1A2B2011084).
Disclosures
The authors declare no conflicts of interest.
References
1. D. Zhao, B. Su, G. Chen, and H. Liao, “360 degree viewable floating autostereoscopic display using integral photography and multiple semitransparent mirrors,” Opt. Express 23(8), 9812–9823 (2015). [CrossRef]
2. S. Maekawa, K. Nitta, and O. Matoba, “Transmissive optical imaging device with micromirror array,” Proc. SPIE 6392, 63920E (2006). [CrossRef]
3. D. Miyazaki, N. Hirano, Y. Maeda, S. Yamamoto, T. Mukai, and S. Maekawa, “Floating volumetric image formation using a dihedral corner reflector array device,” Appl. Opt. 52(1), A281–A289 (2013). [CrossRef]
4. R. Kujime, S. Suyama, and H. Yamamoto, “Different aerial image formation into two directions by crossed-mirror array,” Opt. Rev. 22(5), 862–867 (2015). [CrossRef]
5. Y. Yahagi, S. Fukushima, S. Sakaguchi, and T. Naemura, “Suppression of floating image degradation using a mechanical vibration of a dihedral corner reflector array,” Opt. Express 28(22), 33145–33156 (2020). [CrossRef]
6. Y. Yoshimizu and E. Iwase, “Radially arranged dihedral corner reflector array for wide viewing angle of floating image without virtual image,” Opt. Express 27(2), 918–927 (2019). [CrossRef]
7. S. Choi, S. Park, and S.-W. Min, “Design of ghost-free floating 3D display with narrow thickness using offset lens and dihedral corner reflector arrays,” Opt. Express 28(10), 15691–15705 (2020). [CrossRef]
8. H. Kim, S.-W. Min, and B. Lee, “Geometrical optic analysis of structural imperfection of retroreflection corner-cubes with nonlinear conjugate gradient method,” Appl. Opt. 47(34), 6453–6469 (2008). [CrossRef]
9. H. Kim and B. Lee, “Optimal design of retroreflection corner-cube sheets by geometric optics analysis,” Opt. Eng. 46(9), 094002 (2007). [CrossRef]
10. H. Yamamoto, Y. Tomiyama, and S. Suyama, “Floating aerial LED signage based on aerial imaging by retroreflection (AIRR),” Opt. Express 22(22), 26919–26924 (2014). [CrossRef]
11. C. Gao, X. Sang, X. Yu, X. Gao, J. Du, B. Liu, L. Liu, and P. Wang, “Design, characterization, and fabrication of 90-degree viewing angle catadioptric retroreflector floating device using in 3D floating light-field display system,” Opt. Express 28(17), 24854–24873 (2020). [CrossRef]
12. M. Otsubo, “Optical Imaging Apparatus and Optical Imaging Method Using the Same,” U.S. Patent 8,702,252 B2, (April 22, 2014).
13. B. Javidi, A. Carnicer, J. Arai, T. Fujii, H. Hua, H. Liao, M. Martinez-Corral, F. Pla, A. Stern, L. Waller, Q. Wang, G. Wetzstein, M. Yamaguchi, and H. Yamamoto, “Roadmap on 3D integral imaging: sensing, processing, and display,” Opt. Express 28(22), 32266–32293 (2020). [CrossRef]
14. J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009). [CrossRef]
