1. Introduction

A multi-view autostereoscopic 3D display (MVA3D) has been considered an alternative to eyewear-aided 3D display owing to benefits of providing motion parallax and eyewear-free operation that elevates viewers convenience [1–7]. Despite its advantages, the MVA3D possesses drawbacks of discontinuous motion parallax and viewers’ visual discomfort and fatigue that stem from disaccord between accommodation and vergence of viewer pupils region [8–10]. One of the possible solutions to these challenges is the use of super multi-view autostreoscopic 3D display, i.e., high-density multi-view autostereoscopic 3D display (HD-MVA3D) that realizes multiple view images within a single pupil. It has been suggested and demonstrated using flat panel displays (FPD) [9, 11, 12], projection type light sources [13–15], or hybrid type of FPD and projection system [16,17]. Such a 3D display system uses the FPD, a parallax barrier (PB) or array of lenticular lenses that are embedded on its front surface [9,11,12]. Geometrical ray optics has been applied to design such a 3D display system [18, 19]. The HD-MVA3D demands that a spatial gap between view images be reduced below a viewer pupil size of typically about 5mm [13]. In addition, the HD-MVA3D that employs high display resolution such as 282 pixel per inch (PPI) (commercialized ultrahigh-density resolution on a 15.6 inch screen) will use a display panel of a subpixel size of about 30μm. This indicates possibility of diffraction effects that come into an important play in designing and fabricating the 3D display. In fact, the optical properties of the 3D display, predicted by geometrical optics based ray tracing simulation, produces significant disagreement with the practically measured ones. Thus diffraction effects must be incorporated in designing such a 3D display. However, few detailed studies about the diffraction effects for the PB based MVA3D or HD-MVA3D have been reported to date.

In this paper, we provide theoretical and experimental studies on the diffraction effects that influence optical properties of the HD-MVA3D display that uses a high-density FPD and the corresponding PB. This study also includes a methodology for designing parameters to maximize the relative ratio of view image brightness with respect to image crosstalk, taking into account diffraction effects. We use the Fresnel number (n_FR) for diffraction effects consideration to determine the aperture size of the PB that optimizes the optical characteristics of the 3D display such as view image brightness and image crosstalk. We find that view image brightness maximizes at n_FR ∼ 0.7 while image crosstalk minimizes at n_FR between 0.4 and 0.5. The compromise between the brightness and crosstalk leads to optimization of the relative magnitude of the brightness to the crosstalk at n_FR between 0.5 and 0.6 in a PB based HD-MVA3D.

2. A geometrical ray optics based design of a conventional PB based MVA3D display and its optical characteristics

Figure 1 illustrates schematic of geometrical ray traces in the PB based MVA3D display. The display parameters such as the PB aperture size (W_PBS), the slit period of a PB (T_PB) and the gap between a display panel and a PB (d) will be given by the geometrical relation as follows:

 

Fig. 1 Schematic of geometrical traces of light rays in a PB based multi-view autostereoscopic 3D display.

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Here the D_vp denotes the interval between centers of two adjacent view images at the viewing distance (L), the W_p the fundamental pixel size (subpixel size in one dimensional display consisting of red, green and blue colors), and the n the number of view images provided by the 3D display. Equation (1) shows that the W_PBS can generally be given as slightly smaller than the W_p. Equations (2) and (3) indicate that the L is irrespective of T_PB while depending on d.

We perform the ray tracing simulation of light through an one-dimensional PB, with the designed parameters to obtain optical characteristics of view images that are distributed along a horizontal direction at the optimum viewing distance (OVD) [18]. We define the PB aperture ratio β as the relative ratio of the PB aperture size used in the simulation with respect to the designed one, W_PBS. We vary the β in the simulation while fixing all other parameters to see its effects on view image optical characteristics. These characteristics include view image brightness, image crosstalk, and the relative magnitude of the brightness with respect to the image crosstalk as a function of β, as shown in Figs. 2(a1), 2(b1), and 2(c).

 

Fig. 2 Optical characteristics of view images as a function of PB aperture ratio β in a PB based multi-view autostereoscopic 3D display (W_p = 30μm, D_vp= 5 mm, OVD = 600 mm) (a1) Normalized image brightness I_nv and image crosstalk CT_av in an one-dimensional PB based 3D display (a2) schematic of an one-dimensional PB used for the simulation results shown in (a1), (b1) I_nv and CT_av in a two-dimensional PB (b2) schematic of a two-dimensional PB used for the simulation results shown in (b1). The tilt angle with respect to the display panel is tan⁻¹(1/3). (C) Relative ratio of image brightness to crosstalk.

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We define a view image crosstalk CT_av at its center as the sum of intensity of unwanted view images with respect to that of the given view image. This can then be expressed as

Figure 2(a1) illustrates the results of the ray tracing simulation based on Fig. 1, assuming an one-dimensional PB shown in Fig. 2(a2). This leads to the maximum brightness of view images with zero crosstalk, at β=1, as shown in Fig. 2(a1). This means that the W_PBS given by Eq. (1) corresponds to an optimum condition for the one-dimensional PB case.

However, as shown in Fig. 2(b1), this is not the case of the two-dimensional PB tilted with respect to the display panel. The two-dimensional PB used in the simulation consists an array of stripe-type rectangle apertures as shown in Fig. 2(b2), which corresponds to an array of stripe-type subpixels for red (R), green (G), and blue (B) colors in a two-dimensional display panel. The two-dimensional PB needs to be tilted with respect to the display panel by the angle θ_PB = tan⁻¹(1/3), for image resolution balance between horizontal and vertical directions. Such tilting of a PB broadens the distribution pattern of view image brightness and thus causes maximum brightness and minimum crosstalk to occur at β ≠ 1. For instance, in Fig. 2(b1), maximum brightness occurs at β ≥ 1.8 while image crosstalk CT_av does not vanish at β ≤ 1 unlike Fig. 2(a1).

For designing the PB based MVA3D display and quantitating its view image quality, we define the relative ratio of view image brightness to image crosstalk, γ as given by

Higher γ represents a better quality of view images in the PB based MVA3D. For example, the case of I_nv = 0.8 and CT_av = 0.3 gives γ ∼ 267 while the case of I_nv = 0.4 and CT_av = 0.1 gives γ ∼ 400 (note that I_nv ≤ 1). Figure 2(c) illustrates the γ as a function of PB aperture ratio. With the β approaching unity from large value, the crosstalk decreases towards zero, thus the γ increases rapidly in both cases of one- and two-dimensional PB. However this is not the case when diffraction effects are included in the numerical simulation nor the case of practical measurement of such optical characteristics. It will be seen in the following section that γ maximizes at a certain value of the β which does not equal unity due to the diffraction effects.

In order to analyze optical characteristics of the MVA3D display, which depend on the β as seen in Figs. 2(a1) and 2(b1), we investigate the intensity distribution of individual view images brightness for the different β. Figure 3 shows the simulation results of intensity distributions of view image brightness vs a horizontal position x at the OVD for the three different β values (0.8, 1.0, 1.2) in the one-dimensional PB based MVA3D display. The simulation results produce quasi-triangular distributions as shown in Fig. 3. However, these distributions turn to quasi-Gaussian ones in the two-dimensional tilted PB case.

 

Fig. 3 Simulation results of image brightness distributions vs horizontal position at the OVD for the one-dimensional PB. Here W_p = 30 μm, D_vp = 5 mm, OVD= 600 mm. (a) β = 0.8, (b) β = 1, (c) β = 1.2.

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For β ≤ 1, the adjacent view image light does not reach beyond the center of the viewing zone of a given view image in the one-dimensional PB case as shown by the red broken lines in Figs. 3(a) and 3(b). On the other hand, the adjacent view image light reaches beyond the center position, for β > 1, in the one-dimensional PB case as shown in Fig. 3(c). In the case of the tilted two-dimensional PB structure which produces quasi-Gaussian shape of intensity distribution unlike the one-dimensional PB case, the adjacent view image light reaches beyond the center position regardless of the β magnitude. In both cases of the PB structure, the full width at half maximum value (FWHM) of individual view image brightness distributions increases with the β, leading to expectation of increased crosstalk.

This is manifested in Figs. 4(a) and 4(b) where the relative magnitude of the FWHM to the D_vp, and CT_av increase as the β increases above unity in both cases. At β < 1, the relative magnitude of the FWHM to the D_vp remains unity while CT_av vanishes in the one-dimensional PB case as shown in Fig. 4(a). In the tilted two-dimensional PB case, both the relative magnitude of the FWHM to the D_vp, and CT_av decrease almost monotonically as the β decreases below unity. This indicates that smaller β benefits the lower CT_av at the cost of their lower brightness. On the contrary, Larger β produces higher brightness of view images (saturated beyond a certain value of the β) at the cost of their higher CT_av. This leads to a compromise between two optical properties, i.e. view images brightness and the CT_av to obtain best performance of the 3D display.

 

Fig. 4 Relative magnitude of full width at half maximum to the D_vp and crosstalk as a function of β in a PB based MVA3D display (W_p = 30 μm, D_vp = 5 mm, OVD= 600 mm) (a) the one-dimensional PB (b) the two-dimensional PB tilted by the angle of tan⁻¹(1/3).

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However we note that diffraction effects must be considered when determining the optimum β. This is because, at smaller β, the diffraction plays more crucial role in distributing view image intensity along the horizontal direction, thus influencing more highly both CT_av and brightness of view images. These diffraction effects will be treated in a great detail and discussed for optimum parameter designing in the sections that follow.

3. Diffraction incorporated design of a PB based MVA3D and its optical characteristics

3.1. Theoretical approach for one-dimensional diffraction in a PB based MVA3D

Figure 5 depicts schematically diffraction of light from a point at a position ξ_s on a subpixel surface of a display panel through a PB aperture to a viewing position at the OVD.

 

Fig. 5 Schematic of diffraction of light through a PB aperture from a point on a subpixel surface for distribution of image brightness intensity in a PB-based MVA3D.

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We assume the mutual incoherence between light emitting different points on a subpixel surface. The diffraction results from interference between light emitting from the same point on a subpixel. The mathematical description based on the Huygens-Fresnel principle [20] is given by,

Equation (6) assumes a spherical wave propagation of light from a point source on a subpixel surface. Light field on a PB aperture can be expressed as $U_1(x_1,{\xi }_s)\left(=\frac{\text{exp}(jk{R}_1)}{R_1}\right)$. Here x₁ denotes the position within a PB aperture.

Superposition of light fields that propagate from ξ_s through various position within the PB aperture then takes place at a position (x₂) on the observation plane as indicated by integration in Eq. (6). In Eq. (7), we square the modulus of the superposed outcome U₂(x₂, ξ_s) to obtain light intensity at the observation plane position x₂, as a result of diffraction of light fields. Intensity obtained from diffraction of light from a single position ξ_s can then be integrated over different positions on a subpixel surface, to achieve total intensity of light at a horizontal position x₂ on an observer plane, as shown in Eq. (8). We assume continuous point light sources on a subpixel surface, and mutual incoherence between point sources as mentioned above.

In this computation of intensity of light at x₂ from a subpixel, we exclude contributions from diffraction of light through the other apertures, i.e. periodically positioned adjacent different apertures. This is due to the fact that inclusion of such contributions turns out to induce negligible change in a qualitative nature of intensity distributions owing to the incoherent properties of the pixel emitting light.

3.2. Fresnel Number in the MVA3D (or HD-MVA3D) display

Figure 6 introduces the Fresnel number n_FR that is useful to estimate diffraction effects of light through a PB aperture in a MVA3D display. Each of the Fresnel zones denoted by U_j where j is a non-negative integer in a two-dimensional PB aperture is the ring-shaped zone, across the thickness of which optical phase changes by π as estimated from an observation point orthogonally separated by L from the aperture center [21]. The n_FR is a number of the Fresnel zones on a PB aperture.

 

Fig. 6 Schematic for introduction to the Fresnel number.

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For instance, n_FR = j means that the two-dimensional PB aperture consists of j zones of a ring shape, i.e., U₀,U₁,··· ,U_j. The n_FR can approximately be derived using the MVA3D display parameters in the Eqs. (9)–(12) as follows:

The condition given by Eq. (1) also leads to an approximation of W_PBS ≅ W_p and thus the β can be given by

This means that, diffraction incorporated design enables one to achieve the same n_FR by increasing the β for the decreased W_p and D_vp. For example, the β should become two-fold as both W_p and D_vp get small by factor of two to maintain the same n_FR. Therefore, if diffraction effects of a PB based MVA3D display (especially, the HD-MVA3D display) are taken into account, optimized PB aperture ratio β is expected to vary for different W_p and D_vp.

3.3. Optical characteristics of a PB based HD-MVA 3D display with diffraction effects included

We compute Eqs. (6)–(8) for the intensity distribution formed on the observation plane through the PB aperture from the nearest pixel among a set of n pixels as shown in Fig. 5. These intensity distributions on the observation plane form viewing zones. We calculate the image crosstalk CT_av for a single view image formed around the center of the subpixel nearest viewing zone (the center separated from the PB aperture center by L) as seen in Fig. 5. For calculation of crosstalk at the point, we calculated the intensity on an observation plane from the nearest pixel, I_view, and that from whole pixels, I_av. Then the image crosstalk for a view image at the center of viewing zone is calculated using Eq. (4). Diffraction effects are incorporated in the simulation by using 1^st Rayleigh-Sommerfeld solution based on the Huygens-Fresnel principle [20]. We also check that use of Fresnel approximation for diffraction effects [20] yields the simulation results approximately same as those given by 1st Rayleigh-Sommerfeld solution within disagreement ≤ 0.5%.

Figures 7(a) and 7(b) show the diffraction incorporated simulation results of the I_nv and CT_av as functions of β and n_FR, respectively. The simulation assumes the OVD is 600 mm and the D_vp is 2.5 mm and 5.0 mm. The results are based on the parameters found in the display unit of a commercialized 15.6 inch ultrahigh-density laptop with its subpixel size of 30 μm. Figure 7(a) shows that the normalized view image brightness, I_nv is optimized at β > 1 for each of the two D_vp used in the simulation, in contrast to the case shown in Fig. 2(a1). In addition, the image crosstalk CT_av increases rapidly as the β decreases below unity, unlike Fig. 2(a1). These differences are attributed to the diffraction effects that turn out to dominate display parameter designing in the PB based HD-MVA3D display. It is also seen that the use of different D_vp needs the different β for optimizing optical characteristics such as I_nv and CT_av. However, when these optical characteristics are plotted vs n_FR, the use of different D_vp finds approximately the same n_FR to optimize the view image intensity and crosstalk.

 

Fig. 7 Optical characteristics for different D_vp as a function of β [(a)], and those as a function of n_FR [(b)].

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Figures 8(a) and 8(b) show the simulation results of optical characteristics (I_nv and CT_av) for different W_p, i.e., 30 μm (ultrahigh-density display) and 60 μm (high-density display) while using the same D_vp of 2.5 mm for both cases. The optical characteristics plotted vs a β as shown in Fig. 8(a) indicates that optimization of I_nv (or CT_av) finds the different β for the different W_p. However, when such optical characteristics are plotted vs n_FR, the optimization of the characteristics finds quite similar n_FR, for different W_p which is similar to the cases shown in Figs. 7(a) and 7(b). This indicates that the optimized n_FR value can be considered characteristic of a given PB-based 3D display system and it may thus be beneficial to use the n_FR for optimization of optical properties of a HD-MVA3D display rather than the β. In addition, the use of the n_FR in the display design with diffraction effects taken into account enhances flexibility in the 3D display parameter determination.

 

Fig. 8 Optical characteristics for different W_p as a function of β [(a)], and those as a function of n_FR [(b)].

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We recall that the design of the 3D display system requires optimization of the display parameters. One of the key parameters is the relative magnitude of view image brightness with respect to image crosstalk, γ as defined in Eq. (5). Figure 9 provides the γ values estimated from the simulation with (blue) and without (red) diffraction effects included for comparison.

 

Fig. 9 Simulation results of the relative magnitude of view Image brightness to its crosstalk. W_p = 30 μm, and D_vp =5 mm. The blue and red solid lines represents results obtained with and without diffraction effects included in the simulation, respectively.

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Here the simulation results assume an one-dimensional PB structure. As shown in Fig. 2(a1), the parameter design by approach of geometrical ray optics without diffraction effects considered include the β = 1 that produces highest brightness and zero crosstalk of the view image. This β corresponds to n_FR ∼ 0.11 whereby the Fraunhofer diffraction comes into an important play and broadens the FWHM of the intensity (brightness) distribution of the view image, thus yielding high crosstalk. This cannot be predicted by simulation based on a purely geometrical ray tracing.

The diffraction effects incorporated simulation finds that the n_FR = 0.52 (corresponding to the β = 2.15) maximizes the as shown in Fig. 9. This reflects a compromise between maximum brightness of view image (n_FR ∼ 0.71, β = 2.5) and minimum crosstalk (n_FR ∼ 0.44, β = 1.98) and suggests how the display parameter can be designed by incorporating the diffraction effects in a PB based HD-MVA3D display system.

To gain qualitative understanding of diffraction effects that come into an play to determine the view image characteristics such as γ, we calculate distributions of intensity of a view image for various n_FR as shown in Figs. 10(a)–10(f). Recall that large PB aperture basically produces large FWHM of the distribution in a purely geometrical sense. However, the Fraunhofer diffraction that becomes important at n_FR as small as 0.1 (Fig. 10(a)), broadens the distribution, thus yielding wider FWHM at smaller n_FR at small enough n_FR. These opposing effects indicate that the FWHM of the distribution can be narrowest at the moderate number of n_FR i.e., 0.44 as shown in Fig. 10(b). With increasing the n_FR above 0.44, the FWHM increases. Meanwhile, the view image intensity increases as the n_FR increases until n_FR ∼ 0.7. For 0.7 < n_FR < 1.2, the view image intensity gradually decreases while the FWHM increases. The view image intensity then saturates at above n_FR = 1.2 (corresponding to β ∼ 3.3 for W_p = 30 μm, D_vp = 5 mm). These characteristics may lead to the finding of an optimum value of the n_FR between 0.4 and 0.7.

We also perform the simulation to look into which n_FR can be chosen to optimizes optical characteristics for various D_vp as shown in Fig. 11(a). The n_FR chosen to maximizes the I_nv remains nearly the same as ∼ 0.7 as the D_vp decreases below 5 mm. On the other hands, the n_FR chosen to minimizes CT_av and therefore maximizes the γ, decreases almost linearly with the D_vp. It is thus obvious that the n_FR can be chosen between 0.4 and 0.7 for optimizing the γ and CT_av of the HD-MVA3D display. However, one needs to check what the optimized values of γ and CT_av are. Figure 11(b) shows those values for various D_vp. It is revealed that the optimized values of the γ (CT_av) are very low (high) for small D_vp. This implies that the PB based HD-MVA3D that aims to use the D_vp < 5 mm encounters diffraction imposed inherent limitations in view image quality and thus needs to be designed via compromising between the view image quality and the D_vp to be used. The simulation results shown in Figs. 10(a)–10(f) enable the view image characteristics such as CT_av and γ to be estimated qualitatively in a viewpoint of diffraction for the PB based 3D display.

 

Fig. 10 Simulated intensity distribution of a view image vs x for various n_FR (Red: without diffraction, blue: with diffraction) in the one-dimensional PB based 3D display.

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Fig. 11 (a) Fresnel number (n_FR) that optimizes I_nv, γ, and CT_av for various D_vp in the one-dimensional PB based 3D display (b) the optimized value of γ and CT_av for various D_vp, in the one-dimensional PB based 3D display.

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4. Experimental results and discussion

4.1. Experimental setup

We fabricated the PB based HD-MVA3D display using a display panel taken from a commercialized laptop computer of a 15.6 inch ultrahigh-density resolution. We fabricated 10 PBs for 13-view 3D display, half of which are for D_vp = 2.51 mm, and the other half D_vp = 5.06 mm. The n_FR produced by the set of PBs for a given D_vps, runs from about 0.25 to a value above 0.8. The subpixel size was 30 μm. We designed the 3D display such that the OVD was 600 mm. Each PB size was (H) 60 mm ×(V)85 mm. We estimated that the number of the subpixels affecting each view image on a viewing zone was 48,000 for a given color while 145,000 for a white color. Table 1 shows the parameters used for fabricating the 3D display. In this case, the β = 1, meaning that a PB aperture size determined by geometrical ray optics based simulation, corresponds to n_FR = 0.056 and 0.114 for two different D_vp = 2.51 mm and 5.06 mm, respectively.

Table 1. Display parameters of the HD-MVA3D for comparing experiment and simulation.

View Table

4.2. Experiment and discussion

Figures 12(a)–12(c) present comparison of optical characteristics between experimental measurement and simulation results. This comparison is made at D_vp = 2.51 mm and wavelength of 550nm (green). The normalized view image intensity I_nv and image crosstalk CT_av as functions of PB aperture ratio β are shown in Fig. 12(a) while those as functions of n_FR are shown in Fig. 12(b). The relative magnitude of view image intensity to crosstalk, γ as a function of n_FR is given in Fig. 12(c). It is seen that measurement results presented by five data points (square boxes) in each graph are in a qualitative agreement with the simulation results. This leads to a fact that diffraction effects must be taken into account in designing the PB based HD-MVA3D display and the optimum PB aperture size to provide the maximized γ can be determined by n_FR ≃ 0.56 in such a 3D display. Note that the experimental measurements show higher crosstalk (143%–196%) than given by the simulation (101%–150%) as shown in Figs. 12(a) and 12(b). These features are reflected in Fig. 12(c) where experimental values of the are lower than the ones obtained by simulation. This mismatch can be understood by studying the evolution pattern of the intensity distribution as a function of x for a single view image, when n_FR varies.

 

Fig. 12 Comparison of optical characteristics between experiment and simulation (W_p = 30 μm, D_vp = 2.51 mm, PB Slanted angle= 0 deg) in the one-dimensional PB based 3D display.

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Figures 13(a)–13(e) illustrate the intensity distribution of a view image as a function of x for various n_FR. The red curves represent the simulation results while blue curves the experimental ones. The FWHM of the measured distribution is similar to that of the calculated curve for each case of n_FR. The measurements, however, show slightly broadened features in both wings of the distribution curve compared to the calculated distribution for each n_FR. These features are in agreement with the measured crosstalk which is slightly higher than that estimated by simulation for each n_FR as seen in Figs. 12(a) and 12(b). This difference in crosstalk between experiment and theoretical simulation is attributed to the following factors. First, the calculation of view image intensity is considered only a PB aperture and 13 subpixel set to simplify the calculation. However, experimental setup includes 48,000 green color subpixels and their corresponding PB apertures. Second, the simulation assumes absence of a medium between the PB and the display panel unlike the actual experimental setup where there is glass substance in the gap. This gap medium causes light refraction and thus changes the view image intensity distributions [21].

 

Fig. 13 The intensity distribution of a single view image vs x at the OVD (600 mm) for different n_FR used in Fig. 12 (red curve: the theoretical simulation, blue curve: experiment) in the one-dimensional PB based 3D display.

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In the case of using a tan⁻¹(1/3) tilted PB for the 3D display, the experimental results show higher crosstalk than the simulation ones, which is similar to the non-tilted one-dimensional PB case. This is due to the increased FWHM of the view intensity distribution for each n_FR as shown in Figs. 14(a) and 14(b). It is also seen that use of the tilted PB always produces higher crosstalk than the non-tilted PB case for n_FR in both simulation and experiment, due to the contribution of the PB tilting to the FWHM increase, as similarly seen in the aforementioned simulation results achieved by geometrical ray tracing without diffraction incorporated. This indicates that the n_FR dependence of optical characteristics such as CT_av in the two-dimensional tilted PB case remains similar to that in the one-dimensional (untilted) PB one in a qualitative manner. Thus, the optimum PB aperture size can be estimated for a given display parameter set of the 3D display that uses the two-dimensional tilted PB by way of simulation that assumes an one-dimensional (untilted) PB, to a good approximation.

 

Fig. 14 The FWHM of the view intensity distribution vs n_FR in the HD-MVA3D display with D_vp = 2.51 mm (a) the non-tilted one-dimensional PB (b) the two-dimensional PB tilted by tan⁻¹(1/3) =18.435 deg).

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We fabricate the HD-MVA3D display with the same parameters as above except D_vp = 5.06 mm. We perform the measurement for view intensity distribution and crosstalk and compare the experimental results with the simulation ones for each n_FR. Similarly to the case of D_vp = 2.51 mm, qualitatively good agreement between experiment and simulation results is achieved as shown in Figs. 15(a) and 15(b). We find the optimum value of at n_FR of about 0.52 in both experiment and simulation results. Note that the maximum value of (∼ 1) shown Fig. 14(c) is higher than that (∼ 0.66) in Fig. 12(c). This reveals that, the 3D display designed/fabricated for smaller D_vp produces the smaller optimum value of the in spite of using the same display subpixels.

 

Fig. 15 Comparison of optical characteristics between experiment and simulation (W_p = 30 μm, D_vp = 5.06 mm, PB Slanted angle=0 deg).

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Up to this point, experiment and simulation are performed at a wavelength of green color light (550 nm) to which human eyes are believed to be more sensitive than any other visible wavelength. However, the fact that the actual display uses white color basically comprising three colors (red, green and blue) and the n_FR varies inversely with wavelength (Eq. (13)), stimulates us to examine wavelength dependence of diffraction effects.

We simplify the study of the color dependence of optical characteristics in the 3D display by representing red, green and blue wavelengths of light source by the three specific wavelengths, i.e. 650 nm, 550 nm, and 450 nm, respectively. Simulation that uses different wavelengths of light source provides the results that the view intensity can be maximized at the n_FR of ∼ 0.7 while the CT_av is minimized at the n_FR of ∼ 0.44 for all the three wavelengths as shown in Figs. 16(a) and 16(c). The optimization of the relative magnitude of the view intensity to the CT_av can be given by a compromise between the view image intensity and CT_av and this determines n_FR. This implies that the PB designing is required to consider color dependence of PB aperture size. This leads to a complicated structure of a PB. However, the fact that the optical characteristics of the 3D display that uses white light source results from the combined effects of using three different colors, leads us to check if the wavelength of the middle color, i.e., green can be used as a representative color wavelength for evaluation of optical characteristics of the white light source 3D display.

 

Fig. 16 Simulation results of optical characteristics for the three different wavelengths of light source.

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As shown in Figs. 17(a) and 17(b), the cases of using green and white light sources find quite similar optical characteristics that include the I_nv and CT_av in a quantitative manner. Here we use white light with an optical filter of ∼ 50 nm pass band centered at 550 nm as a green light source. In addition, the simulation that uses a monochromatic green light at 550 nm shows qualitative agreement with the experimental results that uses white light source. This signifies that a single wavelength of green light can be used in simulation to estimate qualitative properties of view image while optical filtered white light (550 nm center) can be used in experiment for quantitative estimation of its optical properties to a good approximation, and thus enabling one to optimize the PB based MVA3D display that uses white light sources being comprised of red, green and blue wavelengths.

 

Fig. 17 Comparison of optical characteristics between the cases of using green and white light sources. (dotted line: simulation results, solid rectangles: experimental ones) (a) as a function of β (b) as a function of n_FR.

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5. Conclusion

The MVA3D display with a PB has routinely been designed by way of geometrical ray tracing. Even numerical simulation for the 3D display evaluation relies on the geometrical approaches. However, it is found that the HD-MVA3D display that adopts a PB of sufficiently narrow aperture, demands different approaches for its designing/fabrication and theoretical evaluation due to diffraction effects that come into significant play in view image quality.

In this study, we present an one-dimensional model to incorporate the diffraction effects for the 3D display designing and evaluation. We examine the optical characteristics as a function of n_FR. Given the W_p and D_vp, this enables the PB slit size to be chosen for the n_FR that optimizes view image quality. It is noted that the display parameter design where D_vp ≤ 5 mm for sufficiently small W_p will require to take the diffraction effects into significant account for optimization of view image quality. Then the consequently optimized slit aperture will give the Fresnel number n_FR < 1. We find that the PB based HD-MVA3D display provides view image brightness which is maximized at n_FR ∼ 0.7 and the CT_av which is minimized at n_FR between 0.4 and 0.5, depending on D_vp. The relative magnitude of the image brightness to the CT_av is maximized at n_FR between 0.5 and 0.6, depending on D_vp. This thus can suggests that the diffraction incorporated design of a HD-MVA3D display that uses a PB (with one-dimension or tilted by tan⁻¹(1/3)) be based on a set of display parameters that satisfy the Fresnel number condition 0.4 < n_FR < 0.7. We also find that, for determining the PB aperture size in the 3D display based on two-dimensional PB that is tilted by a angle required by resolution balance, simulation can be employed with an one-dimensional (untilted) PB assumed. This is because qualitative nature of diffraction modified optical characteristics remains similar between the tilted two-dimensional case and the one dimensional (untilted) one.

Further works may cover experiments that show the detailed dependence of display characteristics on the display parameters including various PB aperture sizes to highlight diffraction effects induced chage in the actual 3D display images. Additional future works can cover diffraction incorporated numerical simulation for the two-dimensional PB based 3D display, effects of the subpixel structure on the view image quality, and diffraction of light from a subpixel through all of the non-nearest PB slits of a given periodicity.