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Backlight system using an interleaved Fresnel lens array that attains a uniform luminance and two-dimensional directional light control

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Abstract

A novel (to the best of our knowledge) Fresnel lens array is proposed to realize a uniform directional backlight with two-dimensional directionality. Autostereoscopic display with the proposed lens array improves image output quality and relieves the viewer’s posture restriction without the need for any additional eye aid. In the proposed lens array, tiny prisms composing two adjacent linear Fresnel lenses are interleaved so that the two lenses may be virtually overlapped and work independently. The widths of the elemental prisms vary depending on the distance from the center of each lens. Thus, the light passing through the two lenses is mixed, which results in higher and more uniform luminance intensity. A prototype of an autostereoscopic display based on the time-multiplexed directional backlight method and made with the proposed lens array attained uniform luminance as well as low cross–talk between left-eye and right-eye images.

© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

Introduction. While a directional backlight can be used for various purposes, such as limiting the viewing zone to protect privacy or saving electric power, its most well-known application is an autostereoscopic display, which does not force the viewer to wear special goggles. By delivering directional light rays to each eye alternatively and synchronizing it with the alternation of right-eye and left-eye images on the LCD panel, a full-resolution stereoscopic image is attained.

One way to attain a directional backlight is to use a thin prism sheet [1,2], although this cannot change the directionality of light to follow the viewer’s position. One of the simplest ways to control the directionality of light is to use a large convex lens or a large concave mirror [35]. This method, however, requires a deep optical distance, which makes the system bulky.

To enable directionality control and thin optics simultaneously, Ishizuka et al. used a convex lens array in place of a large aperture lens [6,7]. Light uniformity is maintained by using a vertical diffuser while aligning the elemental lenses so that they may be shifted in the horizontal direction by different offsets in each row [810]. A couple of improvements have been made to maintain high directionality of the light by solving the problem of aberration [11,12]. However, the use of a vertical diffuser limits the directionality to only one direction.

The authors have recently tried to attain a directional backlight with two-dimensional direction control [13], but we did not achieve this while maintaining enough uniformity of luminance. In the present Letter, we propose a novel lens array to realize uniform luminance and two-dimensional control of directional light at the same time.

Conventional research. The basic principle of an autostereoscopic display using a time-division multiplexing directional backlight composed of a lens array is shown in Fig. 1. The interval between the dot matrix light source and the lens array is equal to the focal length of the elemental lenses so that the directional light may be collimated. By emitting light at the position where the line connecting the observer's eye and the lens center intersects the backlight surface, directional light rays are delivered to each eye of the observer. When the alternation of directional backlight to the left eye and the right eye is synchronized with that of the images on the LCD panel, autostereoscopy is attained.

 figure: Fig. 1.

Fig. 1. Principle of the autostereoscopic display based on time-multiplexed directional backlight using a lens array.

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In this system, more than one viewer can observe the stereoscopic image simultaneously by increasing the number of light sources on the light-emitting plane behind.

However, the seams of the elemental lenses in the lens array are so distinct that the observer can see the shapes of the lenses, which leads to poor image quality in this system. In other words, the intensity of the backlight is not uniform. The cause of this is that the light passing through the peripheral parts of the elemental lenses is weaker than that passing through the centers of the lenses. To improve the uniformity of luminance, Ishizuka et al. placed a vertical diffuser behind the LCD panel while small rectangular lenses were placed with stepwise phase shifts [8,9], which mixed the dark and bright parts of the lenses. A decentered lens array [11] and a curved lens array [12] have also been tested; both of these enhance the directionality of the light by suppressing the effect of the field curvature. A vertical diffuser, however, limits the control of light directionality to only the horizontal direction.

A convex lens array is also used to attain autostereoscopy based on integral imaging. To obscure the distinct seam of the lenses in the coarse integral volumetric imaging display [1419], Kakeya et al. proposed a method that used a unique linear Fresnel lens where small elemental prisms for the left lens and the right lens were interleaved in the connecting part of the lenses, as shown in Fig. 2 [20]. This makes the seam of the adjacent parts of the lenses less distinct. By using two layers of these lenses, a convex lens array with non-distinct seams can be realized.

 figure: Fig. 2.

Fig. 2. Linear Fresnel lens with a non-distinct seam. (above) Standard Fresnel elemental lenses aligned side by side and (below) elemental lenses with interleaved grooves in the connecting part.

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This interleaved structure has also been used to realize a uniform directional backlight with two-dimensional control of the light direction [13]. The structure of the lens array used for this purpose is shown in Fig. 3. Though the basic principle of the lens to obscure the seam is the same as that in Fig. 2, the prisms are interleaved not only in the peripheral parts of the elemental lenses but also in the central parts of the lenses, which means that the elemental prisms are interleaved across the whole Fresnel lens.

 figure: Fig. 3.

Fig. 3. The conventional version of an interleaved linear Fresnel lens.

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As shown in Fig. 4, when two layers of the novel linear lens (four elemental lenses that overlap in the horizontal and vertical directions while one elemental lens is highlighted) are stacked orthogonally [Fig. 4(a)], it is optically equivalent to a stack of overlapping cylindrical lenses in an array [Fig. 4(b)] or a layer of overlapping convex lenses in an array [Fig. 4(c)]. By applying this lens array, the uniformity of luminance is expected to increase without needing to add a vertical diffuser.

 figure: Fig. 4.

Fig. 4. (a)–(c) Equivalent structures of the lens array.

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However, the uniformity of directional backlight when using the lens array in Fig. 4(a) is not sufficient for use in an autostereoscopic display with high image quality.

Proposed method. The problem with the lens in Fig. 3 is the discontinuity of the lenses in the center, where the fragments of steep prisms are facing opposite directions, causing a sudden change of luminance. One way to solve this problem is to gradually shorten the widths of the steeper prisms in order to suppress the effect. A partial cross-section of a conventional linear Fresnel lens with an interleaved structure is shown in the upper part of Fig. 5. The light passing through the prism whose tilt angle is steep (${\alpha _r}$) is weaker than the light passing through the prism whose tilt angle is shallow (${\alpha _l}$), while the horizontal widths of the elemental prisms are all equal. In addition, as shown by the dotted circle in Fig. 5, sudden height drops occur in the connecting part of the elemental prisms.

 figure: Fig. 5.

Fig. 5. Improved redesign of the conventional linear Fresnel lens.

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To solve this problem, we propose an improved design of the interleaved linear Fresnel lens. As shown in the lower part of Fig. 5, the facet of the prism whose tilt angle ${\alpha _l}$ is smaller than that (${\alpha _r}$) of the other prism in the prism pair is extended to intersect the facet of the other prism, as the dotted lines show. Then, the width ratios of the prisms to the total width of the prism pair are rearranged depending on the intersection point so that the width of the steeper prism becomes shorter. Thus, better continuity of the light luminance passing through the lens is expected. Also, this makes the brighter part of the lens wider, which is expected to contribute to the increase in overall luminance. The structure of the lens also becomes simpler as the sudden vertical drops in the connecting part of the two prism segments are removed.

The width of each prism segment is easily calculated from a trigonometric function according to Fig. 5. If the total width of prism pairs is w, and the tilt angles of the prisms are ${\alpha _l}$ and ${\alpha _r}$, respectively, then the widths of two elemental prisms ${w_l}$ and ${w_r}$ are given by

$$w_l = \frac{h}{{\tan}{\alpha_l}},$$
$$w_r = \frac{h}{\tan \alpha_r},$$
where h is the height. Since $w = \; {w_l} + {w_r}$, h is given by
$$h = \frac{w \tan \alpha_l \tan \alpha_r}{{\tan \alpha_l}+{\tan \alpha_r}}.$$
By substituting Eq. (3) into Eqs. (1) and (2), we obtain
$$w_l = \frac{w \tan \alpha_r}{\tan \alpha_l + \tan \alpha_r},$$
$$w_r = \frac{w \tan \alpha_l}{\tan \alpha_l + \tan\alpha_r}.$$

Thus, the ratio of ${w_l}$ to ${w_r}$ is given by

$$\frac{w_l}{w_r} = \frac{tan \alpha_r}{\tan \alpha_l}. $$

Experiment and results. Based on the principle explained in the previous section, we designed an interleaved linear Fresnel lens whose width was 30 mm and focal length was 100 mm, as shown in Fig. 6. The width of the elemental prism pair was 0.6 mm. Note that the width of a unit lens was virtually 60 mm because of the overlap. We made a metal mold based on this design, and injected plastics to produce the elemental lenses. PMMA was selected as the plastic material because it does not affect light polarization.

 figure: Fig. 6.

Fig. 6. Design of the interleaved linear Fresnel lens (values in millimeters).

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We made a prototype autostereoscopic display system that used two layers of the above lens array stacked orthogonally and tested whether it worked as expected. A picture of the prototype system is shown in Fig. 7. The 27-inch LCD panel we used for the prototype system had a resolution of 1920 × 1080 and a 240 Hz maximum refresh rate. A dot matrix light source was simulated by combining an LED surface light and a 27-inch LCD panel that synchronized with the LCD in front at the same refresh rate. Two layers of the novel Fresnel lens array were stacked orthogonally to compose the lens array. The size of the lens array was 540 mm (W) × 300 mm (H), with 18 elemental lenses in the horizontal direction and 10 elemental lenses in the vertical direction.

 figure: Fig. 7.

Fig. 7. Optical layout of the proposed system and the prototype autostereoscopic display system.

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We evaluated the homogeneity of luminance in the proposed system. The angular distribution of the light exiting from the LCD panel placed behind the lens array is shown in Fig. 8(a). We used a TOPCON BM-7AC luminance colorimeter throughout the whole experiment. First, the sRGB gray level given by a digital camera (Panasonic DMC-FZH1) and its corresponding luminance were measured in the experimental environment while displaying a pure gray color, and the polynomial trendline of the corresponding value was found, as shown in Fig. 8(b).

 figure: Fig. 8.

Fig. 8. Angular distribution of the luminance exiting from the LCD placed behind the lens array. (a) Correspondence between the sRGB gray level and (b) the luminance.

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When using the conventional interleaved lens array [12] and the proposed lens array, the central part of the display was captured at 640 × 480 resolution with a digital camera, as shown on the left side of Fig. 9. Histograms showing the gray level distributions given by the conventional lens array and the proposed lens array are shown in Fig. 9(x). The horizontal axis of the histogram represents the pixel gray-level intensity, ranging from 0 to 255, while the vertical axis represents the number of pixels with a particular gray-level intensity. The corresponding luminance distributions, obtained using Fig. 8(b), are shown in Fig. 9(y). As Table 1 shows, the average intensity is increased and the standard deviation of the intensity is decreased by using the proposed lens array.

 figure: Fig. 9.

Fig. 9. Intensity distribution of the captured image. Photos of (a) the conventional method and (b) the proposed method on the left. (x) Gray level distributions and (y) luminance distributions on the right, where curves a and b correspond to the distributions in the images on the left side, respectively.

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Tables Icon

Table 1. Statistical Data for the Intensity Distributions

Next, we used the luminance colorimeter to measure the cross-talk level. To evaluate the cross-talk level, the luminance was measured under the following three conditions: view 0 (both the left-eye image and the right-eye image were black; this was done to measure the ambient luminance); view 1 (the left-eye image was white and the right-eye image was black); view 2 (the left-eye image was black and the right-eye image was white). We set the original point of measurement to the center of the display panel, while the distance was 800 mm away from the panel. The luminance was measured every time the viewing zones for the eyes were moved 1 cm horizontally. The distance of the viewing zones from the original point is denoted $\; d$. In the experiment, the size of the bright area corresponding to each viewpoint was 6 × 6 mm. The bright areas for the left eye and the right eye were separated so that the interpupil distance (PD) could be 65 mm. The results of the experiments with the conventional system and the proposed system are shown in Fig. 10.

 figure: Fig. 10.

Fig. 10. Results of the cross-talk measurements: (a) conventional method and (b) proposed method.

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The suitable eye position for stereoscopy is that where the cross–talk is minimized while the PD is restricted to 6 or 7 cm. We select the data at $d ={-} 5$ cm and $d = 1$ cm as the luminances for the left-eye position and the right-eye position, respectively, so that the cross-talk level is minimized when the PD is 6 cm, while $d ={-} 5$ cm and $d = 2$ cm are chosen when the PD is 7 cm. The observed cross-talk level X is defined by

$$X = \frac{1}{2}\left( \frac{L_{v2} - B_{l}}{L_{v1} - B_{l}} + \frac{R_{v1} - B_{r}}{R_{v2} - B_{r}} \right),$$
where ${L_{v1}}$ and ${L_{v2}}$ are the luminances at the left-eye position under the view 1 and view 2 conditions, ${R_{v1}}$ and ${R_{v2}}$ are the luminances at the right-eye position under the view 1 and view 2 conditions, and ${B_l}$ and ${B_r}$ are the ambient luminances at the left-eye and right-eye positions. By substituting the experimental data into the equation above, the cross-talk level is calculated as 6.5% (PD = 6 cm) or 5.8% (PD = 7 cm). The cross-talk level of the conventional autostereoscopic system is 9.2% (PD = 6 cm) or 8.3% (PD = 7 cm), which means that the cross-talk level given by the proposed method is lower than that given by the previous method.

Conclusion. A novel Fresnel lens array that realizes a uniform directional backlight is proposed. In the proposed lens array, tiny prisms composing two adjacent linear Fresnel lenses are interleaved so that the two lenses may be virtually overlapped. To attain uniform light intensity, the width ratios of the prisms are rearranged such that the width of the steeper prism becomes shorter. Thus, better continuity of the light luminance through the lens is attained. We made a prototype lens array and found that the average intensity was increased and the variance of intensity was decreased, as expected. The autostereoscopic display that uses the proposed lens array attains a lower cross-talk level than that achieved using the conventional lens array.

Funding

Core Research for Evolutional Science and Technology (JPMJCR18A2); Japan Society for the Promotion of Science (17H00750).

Disclosures

The authors declare no conflicts of interest.

Data availability

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

REFERENCES

1. J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

2. M. J. Sykora, J. C. Schultz, and R. L. Brott, Proc. SPIE 16, 78630V (2011). [CrossRef]  

3. T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999). [CrossRef]  

4. A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010). [CrossRef]  

5. H. Kakeya and Y. Arakawa, SIGGRAPH 2000 Conf. Abstract and Applications, p. 178 (2000).

6. S. Ishizuka and H. Kakeya, SID 13 Digest, pp. 1173–1176 (2013).

7. S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014). [CrossRef]  

8. S. Ishizuka, T. Mukai, and H. Kakeya, Proc. IDW’14, pp. 836–839 (2014).

9. S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015). [CrossRef]  

10. T. Mukai and H. Kakeya, Proc. SPIE 9391, 939111 (2015). [CrossRef]  

11. G. Borjigin and H. Kakeya, in Digital Holography and Three-Dimensional Imaging 2019, OSA Technical Digest (Optical Society of America, 2019), paper W2A.2.

12. G. Borjigin and H. Kakeya, Proc. IDW’19, pp. 91 (2019).

13. G. Borjigin and H. Kakeya, Proc. IDW’20, pp. 482 (2020).

14. H. Kakeya, Proc. SPIE 6803, 680317 (2008). [CrossRef]  

15. H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010). [CrossRef]  

16. H. Kakeya, Opt. Express 19, 20395 (2011).  [CrossRef]  

17. H. Kakeya, S. Sawada, Y. Ueda, and T. Kurokawa, Opt. Express 20, 1963 (2012). [CrossRef]  

18. S. Sawada and H. Kakeya, J. Electron. Imaging 21, 011004 (2012). [CrossRef]  

19. S. Sawada and H. Kakeya, Opt. Express 20, 25902 (2012). [CrossRef]  

20. H. Kakeya and S. Sawada, Opt. Lett. 40, 5698 (2015). [CrossRef]  

References

  • View by:

  1. J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).
  2. M. J. Sykora, J. C. Schultz, and R. L. Brott, Proc. SPIE 16, 78630V (2011).
    [Crossref]
  3. T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
    [Crossref]
  4. A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
    [Crossref]
  5. H. Kakeya and Y. Arakawa, SIGGRAPH 2000 Conf. Abstract and Applications, p. 178 (2000).
  6. S. Ishizuka and H. Kakeya, SID 13 Digest, pp. 1173–1176 (2013).
  7. S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014).
    [Crossref]
  8. S. Ishizuka, T. Mukai, and H. Kakeya, Proc. IDW’14, pp. 836–839 (2014).
  9. S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015).
    [Crossref]
  10. T. Mukai and H. Kakeya, Proc. SPIE 9391, 939111 (2015).
    [Crossref]
  11. G. Borjigin and H. Kakeya, in Digital Holography and Three-Dimensional Imaging 2019, OSA Technical Digest (Optical Society of America, 2019), paper W2A.2.
  12. G. Borjigin and H. Kakeya, Proc. IDW’19, pp. 91 (2019).
  13. G. Borjigin and H. Kakeya, Proc. IDW’20, pp. 482 (2020).
  14. H. Kakeya, Proc. SPIE 6803, 680317 (2008).
    [Crossref]
  15. H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010).
    [Crossref]
  16. H. Kakeya, Opt. Express 19, 20395 (2011).
    [Crossref]
  17. H. Kakeya, S. Sawada, Y. Ueda, and T. Kurokawa, Opt. Express 20, 1963 (2012).
    [Crossref]
  18. S. Sawada and H. Kakeya, J. Electron. Imaging 21, 011004 (2012).
    [Crossref]
  19. S. Sawada and H. Kakeya, Opt. Express 20, 25902 (2012).
    [Crossref]
  20. H. Kakeya and S. Sawada, Opt. Lett. 40, 5698 (2015).
    [Crossref]

2015 (3)

S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015).
[Crossref]

T. Mukai and H. Kakeya, Proc. SPIE 9391, 939111 (2015).
[Crossref]

H. Kakeya and S. Sawada, Opt. Lett. 40, 5698 (2015).
[Crossref]

2014 (1)

S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014).
[Crossref]

2012 (3)

2011 (2)

M. J. Sykora, J. C. Schultz, and R. L. Brott, Proc. SPIE 16, 78630V (2011).
[Crossref]

H. Kakeya, Opt. Express 19, 20395 (2011).
[Crossref]

2010 (2)

H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010).
[Crossref]

A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
[Crossref]

2008 (1)

H. Kakeya, Proc. SPIE 6803, 680317 (2008).
[Crossref]

1999 (1)

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Arakawa, Y.

H. Kakeya and Y. Arakawa, SIGGRAPH 2000 Conf. Abstract and Applications, p. 178 (2000).

Borjigin, G.

G. Borjigin and H. Kakeya, in Digital Holography and Three-Dimensional Imaging 2019, OSA Technical Digest (Optical Society of America, 2019), paper W2A.2.

G. Borjigin and H. Kakeya, Proc. IDW’19, pp. 91 (2019).

G. Borjigin and H. Kakeya, Proc. IDW’20, pp. 482 (2020).

Brott, R. L.

M. J. Sykora, J. C. Schultz, and R. L. Brott, Proc. SPIE 16, 78630V (2011).
[Crossref]

J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

Bryan, W.

J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

Fukamib, T.

J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

Hattori, T.

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Hayashi, A.

A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
[Crossref]

Ishigaki, T.

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Ishiguchi, T.

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Ishizuka, S.

S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015).
[Crossref]

S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014).
[Crossref]

S. Ishizuka and H. Kakeya, SID 13 Digest, pp. 1173–1176 (2013).

S. Ishizuka, T. Mukai, and H. Kakeya, Proc. IDW’14, pp. 836–839 (2014).

Ito, H.

A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
[Crossref]

Kakeya, H.

S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015).
[Crossref]

T. Mukai and H. Kakeya, Proc. SPIE 9391, 939111 (2015).
[Crossref]

H. Kakeya and S. Sawada, Opt. Lett. 40, 5698 (2015).
[Crossref]

S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014).
[Crossref]

S. Sawada and H. Kakeya, Opt. Express 20, 25902 (2012).
[Crossref]

H. Kakeya, S. Sawada, Y. Ueda, and T. Kurokawa, Opt. Express 20, 1963 (2012).
[Crossref]

S. Sawada and H. Kakeya, J. Electron. Imaging 21, 011004 (2012).
[Crossref]

H. Kakeya, Opt. Express 19, 20395 (2011).
[Crossref]

H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010).
[Crossref]

H. Kakeya, Proc. SPIE 6803, 680317 (2008).
[Crossref]

G. Borjigin and H. Kakeya, Proc. IDW’20, pp. 482 (2020).

G. Borjigin and H. Kakeya, Proc. IDW’19, pp. 91 (2019).

G. Borjigin and H. Kakeya, in Digital Holography and Three-Dimensional Imaging 2019, OSA Technical Digest (Optical Society of America, 2019), paper W2A.2.

S. Ishizuka, T. Mukai, and H. Kakeya, Proc. IDW’14, pp. 836–839 (2014).

H. Kakeya and Y. Arakawa, SIGGRAPH 2000 Conf. Abstract and Applications, p. 178 (2000).

S. Ishizuka and H. Kakeya, SID 13 Digest, pp. 1173–1176 (2013).

Kobayashi, H.

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Kometani, T.

A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
[Crossref]

Kurokawa, T.

H. Kakeya, S. Sawada, Y. Ueda, and T. Kurokawa, Opt. Express 20, 1963 (2012).
[Crossref]

H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010).
[Crossref]

Mano, Y.

H. Kakeya, T. Kurokawa, and Y. Mano, Proc. SPIE 7524, 752411 (2010).
[Crossref]

Mukai, T.

T. Mukai and H. Kakeya, Proc. SPIE 9391, 939111 (2015).
[Crossref]

S. Ishizuka, T. Mukai, and H. Kakeya, Trans. Inst. Electron., Inf. Commun. Eng., Sect. E E98.C, 1023 (2015).
[Crossref]

S. Ishizuka, T. Mukai, and H. Kakeya, J. Electron. Imaging 23, 011002 (2014).
[Crossref]

S. Ishizuka, T. Mukai, and H. Kakeya, Proc. IDW’14, pp. 836–839 (2014).

Nakao, K.

J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

Sakai, A.

A. Hayashi, T. Kometani, A. Sakai, and H. Ito, J. Soc. Inf. Disp. 18, 507 (2010).
[Crossref]

Sawada, S.

Sawaki, A.

T. Hattori, T. Ishigaki, K. Shimamoto, A. Sawaki, T. Ishiguchi, and H. Kobayashi, Proc. SPIE 3639, 66 (1999).
[Crossref]

Schultz, J. C.

M. J. Sykora, J. C. Schultz, and R. L. Brott, Proc. SPIE 16, 78630V (2011).
[Crossref]

J. C. Schultz, R. L. Brott, M. Sykora, W. Bryan, T. Fukamib, K. Nakao, and A. Takimoto, SID 09 Digest, pp. 127 (2009).

Shimamoto, K.

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Data availability

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

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Figures (10)

Fig. 1.
Fig. 1. Principle of the autostereoscopic display based on time-multiplexed directional backlight using a lens array.
Fig. 2.
Fig. 2. Linear Fresnel lens with a non-distinct seam. (above) Standard Fresnel elemental lenses aligned side by side and (below) elemental lenses with interleaved grooves in the connecting part.
Fig. 3.
Fig. 3. The conventional version of an interleaved linear Fresnel lens.
Fig. 4.
Fig. 4. (a)–(c) Equivalent structures of the lens array.
Fig. 5.
Fig. 5. Improved redesign of the conventional linear Fresnel lens.
Fig. 6.
Fig. 6. Design of the interleaved linear Fresnel lens (values in millimeters).
Fig. 7.
Fig. 7. Optical layout of the proposed system and the prototype autostereoscopic display system.
Fig. 8.
Fig. 8. Angular distribution of the luminance exiting from the LCD placed behind the lens array. (a) Correspondence between the sRGB gray level and (b) the luminance.
Fig. 9.
Fig. 9. Intensity distribution of the captured image. Photos of (a) the conventional method and (b) the proposed method on the left. (x) Gray level distributions and (y) luminance distributions on the right, where curves a and b correspond to the distributions in the images on the left side, respectively.
Fig. 10.
Fig. 10. Results of the cross-talk measurements: (a) conventional method and (b) proposed method.

Tables (1)

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Table 1. Statistical Data for the Intensity Distributions

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

w l = h tan α l ,
w r = h tan α r ,
h = w tan α l tan α r tan α l + tan α r .
w l = w tan α r tan α l + tan α r ,
w r = w tan α l tan α l + tan α r .
w l w r = t a n α r tan α l .
X = 1 2 ( L v 2 B l L v 1 B l + R v 1 B r R v 2 B r ) ,

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