Abstract

A three-dimensional simulation model calculating the optical intensity distribution for the entire screen of an autostereoscopic display at a given eye position was developed in this study. A parallax barrier array was used for the optical model and reverse ray tracing of light from the observer’s eye to the subpixels through the slits of the barrier was performed based on reverse geometrical optics. By investigating the optical behavior of the displayed image for the nine-view design condition for various viewing distances, we found the inhomogeneous crosstalk from the unwanted views and predicted segmented images which were comprised of multiple images from different views on the entire display screen. From the results, our simulation model shows good potentiality for predicting the displayed image on the entire display screen of autostereoscopic displays for various positions of the observer’s eye with sufficient calculation speed.

© 2015 Optical Society of America

1. Introduction

Autostereoscopic display technologies have been developed in various ways, such as by using spatial or time multiplexing technologies. Among them, spatial multiplexing technologies with simple optical filters attached in front of the display panel can be promising candidates to prepare the three-dimensional (3D) display industry of the next generation owing to their cost effective structure and viewing comfort [1, 2].

For spatial multiplexing technologies, the autostereoscopic display is composed of an optical filter, such as a parallax barrier or a lenticular array, and an image display panel. The parallax barrier or lenticular array is comprised of slits or lenticular arrays with their designed pitches. The image display panel is comprised of a number of pixels. Each pixel has three subpixels, which are corresponded to red, green and blue color components of a single pixel of display panel. The subpixels are assigned to each view of the different parallax disparities, and grouped by the number of parallax images which can be defined as the number of views [3]. The types of autostereoscopic displays which have multiple images in various directions are, in general, called multiview autostereoscopic displays. Due to this optical structure, the parallax barrier or lenticular array attached in front of a subpixel group transmits images for viewing which have parallax disparities in different directions by filtering the light from each subpixel assigned to each image of view.

Based on the optical relationships between the optical filter and the subpixel array, the pitches of the parallax barrier or lenticular array are designed for a viewer’s eye at a given position to see all the subpixels of a given view over the entire display area, thereby leading to an optimal viewing distance from where the viewer watches the desired image on the entire display screen of the autostereoscopic display. However, even for the designed pitches of the parallax barrier or lenticular array, the autostereoscopic display can produce some mixed images among the views, resulting in a non-uniform image over the entire display screen when the viewer’s position departs from the optimal viewing distance, and we can guess that the image quality over the entire screen of the autostereoscopic display depends on the viewer’s distance. Therefore, it is an essential problem to predict the optical behavior for the entire display screen in accordance with the observer’s position by numerical simulation before fabricating autostereoscopic displays, since it requires a lot of effort in time and cost to fabricate the autostereoscopic displays with high precision.

Several simulation models describing the optical phenomena for the barrier or lenticular array of an autostereoscopic display were introduced by [4–9]. These reports are highly focused on the calculation of the angular distribution of the light profile at a local point of the display screen. Another method based on the diffractive optics was introduced by [10]. This method also describes the calculation of the angular distribution of the light profile at a local point of the display screen and a conceptual idea of reverse optics was introduced in this paper. However, with the above simulation model, sophisticated optical characteristics such as the luminance and the 3D crosstalk at a local point on the display can be obtained, but it requires too much effort in simulation time and memory in order to predict the actual image quality over the entire display screen. Therefore, an optical model for effective and direct simulation of the displayed image over the entire screen of the autostereoscopic display is strongly required.

In this study, we propose a 3D simulation model calculating the displayed image over the entire screen of the autostereoscopic display in order to predict the actual image quality for the given observer’s position. We use the parallax barrier array in formulating the optical model, since the barrier model is simpler than the lenticular model and they have similar optical functions in autostereoscopic displays even though they have slightly different optical structures. By introducing the observer’s position and the points at the center of the slits between the barriers, the reverse ray tracing of light from the observer to the subpixels through the points at the center of the slits between the barriers is performed based on Snell’s law of refraction. Based on our 3D model, the intensity distributions over the entire screen for all the different views at the various viewing distances will be analyzed, and the variations in the optical characteristics of the display screen in accordance with the viewing distance will be also discussed.

2. Basic concept of modeling

In order to obtain the intensity distribution over the entire screen of the autostereoscopic display, we first locate the observer’s eye at E = (xe, ye, ze) in front of the display panel, and the display surface is laid on the xy-plane at z = 0. The optical configuration for the 3D model is illustrated in Fig. 1. We assume that the parallax barrier is attached on a flat panel display with a physical gap g from the subpixel array of display panel, as shown in the figure. The interfacing surface between the two refractive media ni and nt is considered to also be located at z = 0 and to be comprised of the parallax barrier array, therefore, the subpixels of the display panel are immersed by the medium of refractive index ni, while the observer is located in free space having the refractive index nt.

 figure: Fig. 1

Fig. 1 Optical configuration of our 3D simulation model calculating the intensity distribution over the entire screen of the autostereoscopic display with the parallax barrier array. The coordinate systems, planes of interface, parallax barrier and the subpixel array are described. The wave vectors of the incident and the transmitted light rays are also depicted in the figure.

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Assume that the light ray emitted from a given starting point S = (xs, ys, zs) within the arbitrary subpixel area is incident to a given crossing point C = (xc, yc, zc) on the interfacing surface and that the light ray is transmitted from the crossing point C and reaches the position of the observer’s eye E, as shown in the figure. The wave vectors of the light rays incident to and transmitted from the crossing point can be expressed as in the following equation:

ki=2πλni(sinθicosφix+sinθisinφiy+cosθiz),
kt=2πλnt(sinθtcosφtx+sinθtsinφty+cosθtz).
Here, λ is the wavelength of the light in free space, and ni and nt are the refractive indices of the glass and the transmitted medium, respectively. The symbols θi, ϕi, θt, and ϕt, are the inclination and azimuthal angles of the incident and transmitted wave vectors at the crossing point on the interfacing surface. The symbols x, y and z denote the unit vectors along the x-, y- and z-axes, in the Cartesian coordinate system. Here, the inclination and azimuthal angles of the transmitted wave vectors at a given crossing point can be expressed as in Eq. (2) with the position of the observer’s eye, E = (xe, ye, ze).
θt=tan1[(xexc)2+(yeyc)2zezc],
φt=tan1[(yeyc)(xexc)].
The inclination and azimuthal angles of the incident wave vector can be obtained from Eq. (3) by Snell’s law of refraction, assuming the interfacing surface to be a plane.
θi=sin1[(nt/ni)sinθt],
φi=φt.
Therefore, the starting point S = (xs, ys, zs) can be obtained from Eq. (4) by the inclination and azimuthal angles of the incident wave and the given crossing points on the display screen for a given observer’s eye position.
xs=tanθicosφi(zczs)+xc,
ys=tanθisinφi(zczs)+yc,
zs=g.
Here, since we assumed that the crossing and starting points are located at the interfacing surface and the plane of the subpixel array, respectively, the zc and zs values are equal to zero and –g, respectively, as shown in Fig. 1.

For the numerical simulation of the intensity distribution on the display screen, we define the calculation domains on the planes of the parallax barrier, subpixel array and the display screen, as illustrated in Fig. 2. First, assume that all the crossing points on the display surface, where the light rays are transmitted, are located along the center lines of the slits as illustrated in Fig. 2(a). Considering that the slits between the parallax barriers, having a slanted angle ϕs, are located on the plane at z = 0 and each slit is numbered by the index l representing the order of each slit, the equation of the center line of the slits can be expressed as follows:

x=tanφsy+Phl+xb.
Here, Ph is the horizontal pitch of the parallax barrier and xb is the horizontal bias of the parallax barrier from the origin of the coordinate system, as illustrated in Fig. 2(a). We also use the calculation points in the y-direction with a calculation step of Δy. The crossing point C = (xc, yc, zc) can then be expressed along the center line of each slit, as expressed in Eq. (6), with the index of the m-th line of calculation and the l-th slit of parallax barrier array.
xc=tanφsyc+Phl+xb,
yc=Δym,
zc=0.
Here, as defined above, xc and yc are confined within the interval of – W/2 ≤ xcW/2 in the horizontal direction and – H/2 ≤ ycH/2 in the vertical direction, where W and H are the width and the height of the display screen of the autostereoscopic display. The calculation step Δy in the vertical direction can be defined as Δy = H/Nc, where Nc is total number of calculation lines in the vertical direction. An integer m is the index of the horizontal line for calculation points from 0 to Nc, as illustrated in Fig. 2(a).

 figure: Fig. 2

Fig. 2 Illustrations of the calculation domains for (a) the parallax barrier array, (b) the subpixel array and (c) the display screen of the autostereoscopic display. The starting point (S) and crossing point (C) of each light ray are introduced. The slanted angle ϕs and horizontal barrier pitch Ph of the barrier array, and the subpixel sizes Px and Py in the x- and y-directions are defined. The rectangular grids on the display screen are also illustrated in the figure.

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The next step is to calculate the starting point S = (xs, ys, zs) on the plane of the subpixel array in order to find out the view number of the subpixel from where the light ray started. As described above, the starting points can be obtained from Eqs. (2) to (4) for a given crossing point and the position of the observer’s eye. Figure 2(b) shows the location of the starting points and the subpixel arrangement to find out the view number assigned to each subpixel. By the assumption that the subpixels are square shaped and are ruled into horizontal and vertical pitches Px and Py, the left-top edge point of each subpixel can be expressed as (xi, yj) = (PxiV/2, H/2 – Pyj) with i and j indicating the indices of the subpixel in the horizontal and vertical directions, respectively. Here, the width, W, and the height, H, of the entire display screen can be expressed as W = PxNx and H = PyNy for the total number of subpixels, Nx, and the total number of subpixel lines, Ny, in the horizontal and vertical directions, respectively. When the starting point is calculated to be located within the subpixel region of PxiV/2 ≤ xs < Px⋅(i + 1) – V/2 in the x-direction and H/2 – Py⋅(j + 1) ≤ ys < H/2 – Pyj in the y-direction, the i-th and j-th indices of the subpixel can be obtained to be integers satisfying Eqs. (7a) and (7b), respectively.

xs/Px+Nx/21<ixs/Px+Nx/2,
ys/Py+Ny/21j<ys/Py+Ny/2.
By obtaining the integers i and j, we can find out the view number of the light ray starting from the given subpixel with the following equation from Ref [9]:
vij=NtMod[i+1(j+1)(Py/Px)tanϕs+xos,Nh](NtNh).
Here, Nt and Nh are the total number of views and the number of horizontal subpixels in the subpixel group, respectively. φs and xos are the intended view map angle and the horizontal off-set value of the subpixel array, respectively, in order to provide the desired view to the observer’s eye by forming the view numbers on the entire subpixels. Mod(α, β) is the remainder when α is divided by β. Therefore, after the starting points (xs, ys, zs) on the plane of the subpixel array are obtained for every crossing point (xc, yc, zc) on the barrier plane, we can finally get the view number information of the starting point to the observer’s eye.

To obtain the intensity distribution of light on the display screen at a given viewing distance, we first set the display screens to have a homogeneous grid spacing given by Δx and Δy and grid numbers of Mx and My in the x- and y-directions, as shown in Fig. 2(c). By solving Eqs. (2), (3) and (4) for the crossing point (xc, yc, zc) at the barrier surface for a given observer’s eye position (xe, ye, ze), we can trace reversely the path of a light ray in order to find out the point (xs, ys, zs) on the plane of the subpixel array. After finding out the view number of the given light ray with its starting point, the intensity of each light ray can then be collected on a square grid corresponding to its crossing point since we assumed the crossing points are located at the display plane. Of course, the optical intensity of each light ray also has to be calculated to collect the intensity of the light rays on the square grid at the display screen.

The optical transmittance of the transmitted light at the crossing point with respect to the incident light can be calculated by Eq. (9) for the cases of transverse magnetic (TM) and transverse electric (TE) waves [11].

T=ntcosθtnicosθi[2nicosθinicosθi+ntcosθt]2,
Tll=ntcosθtnicosθi[2nicosθinicosθt+ntcosθi]2.
Considering the incident light to be unpolarized, the intensity of the transmitted light It(θt, ϕt) can be obtained by averaging the transmittances of the TM and TE waves with the intensity of the incident light Ii(θi, ϕi) for the given inclination and azimuthal angles.

It(θt,φt)=12(T+Tll)Ii(θi,φi).

3. Numerical simulation

The simulation procedure is illustrated in Fig. 3. Preparing the 3D simulation, we set the position of the observer’s eye (xe, ye, ze) and the parameters for mapping the view numbers on to the complete set of subpixels of the panel. Here, the total number of views, Nt, the horizontal number of subpixels, Nh, in the subpixel group comprised of all the subpixels assigned to all views, the intended view map angle, φs, and the horizontal off-set value, xos, are the design parameters forming the view map for the panel having the horizontal and vertical number of subpixels, Nx and Ny, with horizontal and vertical subpixel pitches, Px and Py. The intensity distribution of the initial light from a subpixel immersed in the refractive media, Ii(θi, ϕi), has to be given for the preparation of the simulation for the more sophisticated simulation of intensity profiles. The refractive indices of the incident and transmitted media, ni and nt, are given as material parameters. The gap between the lenticular bottom and the subpixel array, g, the horizontal barrier pitch, Ph, and the slanted angles, ϕs, are also given as design parameters for the barrier array. The numbers of grid lines in the horizontal and vertical directions, Mx and My, on the display screen position must be defined for the post-processing of the intensity distribution by collecting the intensities of the light rays for the given square grids corresponding to the light rays. Finally, the number of crossing points along the center lines of slits between the barriers, Nc, and the number of slits over the entire display screen, Ns, are given for the numerical calculation. After all the information regarding the numerical and design parameters is determined, the simulation can be performed via the following procedures.

 figure: Fig. 3

Fig. 3 Simulation procedure for obtaining the intensity distribution over the entire display screen. After configuring all the numerical and physical parameters, reverse tracing for each light ray from the starting point to the observer’s eye is performed. All the intensities of the individual light rays are collected at each grid point corresponding to their position in relation to the crossing points on the display screen.

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First, for the m-th and l-th crossing point, C, which is given by Eq. (6), the inclination and azimuthal angles, θt and ϕt, of transmitted light ray can be obtained by Eq. (2) for the observer’s position, E. Here, the inclination and azimuthal angles, θi and ϕi, of the light incident to the crossing points can be obtained by Eq. (3) using Snell’s law of refraction. By calculating Eq. (4), which is the relation between the starting point and the crossing point, the starting point of the incident light ray can then be obtained. After the starting point is calculated, we can find out the subpixel from which the light ray started by calculating the horizontal and vertical indices, i and j, from Eq. (7). From the view number of the subpixel given by Eq. (8) with its horizontal and vertical indices of i and j, we can record the view number of the light ray for calculating the intensity distributions of the given views on the display screen. Finally, the intensity, It, of the transmitted light ray is calculated with Eq. (10) and collected within the square grid on the display screen where the crossing point of the light ray belongs. We can then obtain the intensity distributions on the display screen under the given structure for multiview autostereoscopic displays by repeating this procedure for the entire set of crossing points for all the centers of the slits between the parallax barriers.

The basic construction of the simulation structure is illustrated in Fig. 4. We used the nine-view design condition commonly used for autostereoscopic displays. The optimal viewing distance is designed to be 2.5 m, with the horizontal intervals between neighboring views being 32.5 mm at the optimal viewing distance. For the given subpixel pitches, the gap between the parallax barrier array and the pixel array is calculated to be 6.0 mm from the common design rule of autostereoscopic displays to have an optimal viewing distance of 2.5 m. We have assumed the average inter-pupil distance for human eyes to be 65 mm for the design of autostereoscopic displays. The resolution of the display panel is ultra high definition (UHD) for which the numbers of subpixels in the horizontal and vertical directions, Nx and Ny, are 11,520 and 2,160, respectively. The subpixel pitches in the horizontal and vertical directions, Px and Py, are taken to be 105 μm and 315 μm, respectively. Here, we neglected the black matrix around the subpixel array for the simplicity of the simulation. Since we used a subpixel group which is comprised of 4.5 and 2 subpixels in the horizontal and vertical directions, respectively, the intended view map angle φs is designed to be tan−1(1/6) = 9.462° with respect to the y-axis, as shown in Fig. 4. The horizontal pitch is obtained to be 471.735 μm to have full white images of given view at the optimal viewing distance. The slanted angle of barrier ϕs is also given to be 9.462°, the same as the intended view map angle of the nine-view condition. As a numerical parameter, the number of slits, Ns, is given to be 2,560, and the number of crossing points in a single slit, Nc, is given to be 108,000 within the calculation domain of the entire display screen. The numbers of grids on the detector screen, Mx and My, in the horizontal and vertical directions are 192 and 108 in order to collect the intensities of the light rays for their corresponding square grid. All the physical and numerical parameters used in the simulation are summarized in Table 1.

 figure: Fig. 4

Fig. 4 The design condition of nine-view autostereoscopic displays used in our simulation. The view map arrangement of subpixels assigned to all the view numbers is illustrated. We used the subpixel group of nine-view autostereoscopic displays having 4.5 subpixels and 2 lines in the horizontal and vertical directions. The intended view map angle of our configuration was designed to be φs = tan−1(1/6) = 9.462° in our configuration of the simulation.

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Tables Icon

Table 1. List of Parameters and Values Used in the Simulation

First, we simulated the intensity distributions over the entire screen of the nine-view autostereoscopic display under nine test patterns corresponding to each view at a given fixed observation point. Figure 5 shows the intensity distributions on the display screen observed at E = (0.0, 0.0, 2.5) m for the test pattern for all the different views. Here, we used the test pattern for a given view with each subpixel having white data for that corresponding view and the other subpixels having black data for the other views. For our design condition, to make a full white image for view 5 show on the entire screen at the center front of the display screen given by the above location E, we set the off-set of the view arrangement xos and the horizontal bias xb of the slits to be −2.8 and 26.25 μm, respectively. Furthermore, since the barrier pitch was designed to be 471.138 μm for the single view image to be watched over the entire display screen at the given observer position, all the subpixels corresponding to view 5 can be seen through all the slits of the barrier array, leading to a full white image over the entire display screen as shown in Fig. 5. For the neighboring views of 4 and 6, since they are intended overlap with the desired view from our design condition, light leakages in views 4 and 6 can also be simultaneously seen over the entire the display screen.

 figure: Fig. 5

Fig. 5 Calculated intensity distribution on the display screen observed at the optimal viewing distance of ze = 2.5 m under various test images for which subpixels corresponding to a given view are white and the others are black. The observer’s eyes are located at the center front of the display, i. e., xe = 0 and ye = 0, where view 5 among the nine views can be observed over the entire display screen. Even though the observer’s eyes are located to see the full white image of view 5, the neighboring images for view numbers 4 and 6 are also observed for almost the whole display screen, and views for 3 and 7 are also found around the corners of the display screen.

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In addition, the light leakages from the second nearest views of 3 and 7 are also found around the top-left or the bottom-right corners of the display screen, respectively, due to the path of the light ray diagonally formed by the geometric relationships between the observer’s eye and the slits at the screen corners, as shown in Fig. 5. Therefore, if we try to eliminate the light leakages from view 3 and view 7 around the corners of the display screen, we must correct the assigned view numbers of their corresponding subpixels by shifting the subpixel groups to proper locations. Of course, from our simulation results, it is found that the images of views 1, 2, 8 and 9 are clearly blocked by the parallax barriers owing to our design condition of the nine-view autostereoscopic display. As shown in the figure, the images are also shown to have symmetries between views 4 and 6 and between views 3 and 7 from point reflection through the origin of the Cartesian coordinate system due to the optical symmetry. From the results, we may guess that the unwanted views can give rise to inhomogeneous ghost images even for the optimal viewing distance due to the optical geometry between the observer’s eye and the optical filter in front of the subpixel array of the autostereoscopic displays. This kind of images is similar to the experimental results presented in the previous paper [12]. Even though the experimental setup in the paper and the design condition of our simulation is not exactly the same, the light distributions of the center and neighboring views from the snapshots and our simulation results are coincide well with each other. Therefore, we think that our simulation model describes well for the calculation of intensity distribution on the entire display screen.

For the numerical simulation, less than 2,560 × 108,000 number of reverse rays corresponding to every crossing point on the display surface were used. In our simulation, it takes less than 10 minutes for all the reverse rays under a computer system configured with Intel® CORETM i5 CPU and 4 Giga-bytes RAM. If we consider the time and effort for the fabrication of autostereoscopic displays, time consumption of less than 10 minutes of our simulation model is sufficient for obtaining the optical characteristics of displayed image on the autostereoscopic displays. Therefore, we think that our simulation model based on the reverse ray tracing optics with analytical forms has sufficiently fast calculation speed to obtain the displayed image under normal design condition of autostereoscopic displays.

Next, we simulated the intensity distribution over the entire display screen of the autostereoscopic display in accordance with the viewing distance of the observer. The intensity distributions on the entire display screen under the test image of view 5 are plotted in Fig. 6 at viewing distances in the range from 0.5 m to 4.5 m in 0.5 m steps. As already seen in Fig. 5, we can see the uniform intensity distribution for the entire display screen at the optimal viewing distance of ze = 2.5 m. However, when the observer departs from the optimal viewing distance, black and white regions are formed in several patterns on the entire display screen, as shown in Fig. 6, displaying only the image of view 5 within the white region. The reason for the forming of the black and white regions on the screen is the optical mismatch between the slits and the subpixels of view 5 when the observer departs from the optimal viewing distance. Here, the patterns of the white regions are formed in different ways for the different positions of the observer’s eye. As shown in the figure, each white pattern is slightly inclined with respect to the vertical direction when the observer departs from the optimal viewing distance, and they are changed to curved lines due to the geometrical relationship between the observer’s eye and the slits on the screen in the diagonal direction.

 figure: Fig. 6

Fig. 6 Calculated intensity distribution over the display screen observed at various viewing distances from z = 0.5 m to 4.5 m with a step size of 0.5 m for a test image for which the subpixels correspond to view 5. The white regions in accordance with the viewing distance are found where the image for view 5 is displayed on the display screen.

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For a viewing distance larger than the optimal viewing distance, the number of white regions increase slowly and the width of the white region decreases from around 40% to 15% of the width of the display panel when the viewing distance changes from 3.0 m to 4.5 m in 0.5 m steps, as shown in Fig. 6. In contrast, for a viewing distance smaller than the optimal viewing distance, the number of white regions abruptly increases from 1 to 11 and the width of the white regions decrease rapidly from 30% of the width of the display panel to very thin lines, when the viewing distance decreases from 2.0 m to 0.5 m with the same step value of 0.5 m. Due to the geometric relationship between the eye and the local point on the display screen, the variation in the viewing angle between the eye and the local point on the display screen at a viewing distance close to the display panel is greater than that for a viewing distance far from the display panel. From this fact, we may guess that the displayed images can change very quickly with the movement of the observer’s position, leading to many image segmentations on the display screen.

To analyze in detail the optical behavior with respect to viewing distance of the displayed images on the screen of the nine-view autostereoscopic display, we simulated the intensity distribution on the screen at around the optimal viewing distance with smaller steps of viewing distance. Figure 7 shows the cross sectional intensity distributions along the horizontal position on the display screen under the test images of views from 1 to 9 at viewing distances from 2.1 m to 2.9 m with a 0.1 m step size. As expected in Figs. 5 and 6, a constant intensity distribution over the entire display screen is observed around at the optimal viewing distance of 2.5 m as plotted in Fig. 7. However, when the observer leaves from the optimal viewing distance, all the intensity distributions are not constant and are similar to a trapezoidal function which has a constant region and a linearly increasing and decreasing region, along the position on the screen in the horizontal direction. Here, the full width at half maximums (FWHMs) of the intensity distributions are obtained to be 30%, 45%, and 100% of the width of the display screen at the viewing distances of 2.1 m, 2.2 m, and 2.3 m, respectively. For these viewing distances, the FWHM of each intensity distribution increases as the viewing distance becomes closer to the optimal viewing distances. The FWHM of each intensity distribution for the viewing distances of 2.6 m, 2.7 m, 2.8 m and 2.9 m are obtained to be 86%, 60%, 44% and 35% of the width of the display screen, respectively, and they become narrower as the viewing distance becomes greater than the optimal viewing distance. Especially at the viewing distances of 2.4 m and 2.5 m, the FWHM is maximized since all the intensity values are around 100% of the maximum, leading to the full image of each view from the left to the right sides of display screen.

 figure: Fig. 7

Fig. 7 Cross sectional intensity distribution on the display screen observed at various viewing distances from z = 2.1 m to 2.9 m with a step size of 0.1 m for various test images of the various views. The FWHM of each intensity distribution increases gradually for viewing distances between 2.1 m and 2.5 m and decreases again for viewing distances greater than 2.5 m.

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Images of the given view mixed with the other views are also found over the entire display screen due to the slanted structure of the parallax barrier for the nine-view configuration as we described above. Especially for the displayed image on the entire screen, we can also find segmented images which are comprised of various views. Looking at Fig. 7 in detail, all the mixed and segmented images of the views are displayed on the entire screen, simultaneously with their own order from the left to the right side of the display screen. When the observer is located far from the optimal viewing distance, the segmented image of each view is shown in forward order from the left to the right sides of the display screen. This phenomenon is different from the case when the observer is located between the display and the optimal viewing distance. In the case when the viewing distance is smaller than the optimal viewing distance, we can observe the segment images of each view in reverse order from the left to the right side of the display screen, as shown in the figure. Therefore, we can imagine that the displayed images are shown to be mixed and segmented for every viewing position except at the optimal viewing distance, and the order of segmented image is in forward order at a viewing distance far from the optimal viewing distance, while the order of the segmented images is reversed at viewing distances between the display and the optimal viewing distance.

Finally, we calculated the positional distribution of the 3D crosstalk in order to analyze the amount of ghost imagery for a single eye for the nine-view autostereoscopic displays. Figure 8 shows the 3D crosstalk distribution in the horizontal direction at y = 0 for various viewing distances from 2.1 m to 2.9 m with a step size of 0.1 m, for which the intensity distributions are obtained in Fig. 7. The 3D crosstalk distributions Xi(x, y) for i-th view are calculated by the following equation [3];

Xi(x,y)=1Li(x,y)[j=1j=NtLj(x,y)Li(x,y)]×100%
where Li (x, y) and Lj (x, y) are the intensity distributions which are obtained by our simulation model for the test pattern of the i-th and j-th views, respectively. Nt is the total number of views as already mentioned above. As shown in the figure, at the viewing distances of ze = 2.4 m and 2.5 m, the 3D crosstalk distribution for view 5 are calculated to be a constant value of around 100% for the entire display screen, since the light amount from view 5 is constant and almost equal to the summation of light leakages from all the other views over the entire display screen. However, at all the viewing distances except ze = 2.3 m, 2.4 m and 2.5 m, the 3D crosstalk increases exponentially as the position on the screen gets further from the center of the screen, due to the linearly increasing or decreasing intensity distribution of view 5 outside the constant intensity region. Especially for ze = 2.3 m, the 3D crosstalk increases linearly as shown in Fig. 8. Here, we can also find out that the widths of the constant 3D crosstalk regions are the same as the widths of the constant intensity region as plotted in Fig. 8. Therefore, we may guess that there must be a transition from the constant to the linear intensity distributions of view 5 for the viewing distances from our simulation results.

 figure: Fig. 8

Fig. 8 Cross sectional 3D crosstalk distributions in the horizontal direction of the display screen under various viewing distances. The 3D crosstalk value around the center of the display screen is 100% for all the viewing distances and increases as the position on the display screen becomes far from the center of the display screen at viewing distances other than the optimal viewing distance.

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Going back to the discussion about the FWHM of each intensity distribution in the case of nine-view autostereoscopic displays, the 3D crosstalk can be obtained to be around 300% at the position of the half maximum of the intensity distribution for view 5 from the simple calculation of Eq. (11). The horizontal size of the display of view 5 at each viewing distance under 300% 3D crosstalk is then also the same as the width of the FWHM for all the cases of different viewing distances. Therefore, if we make a restriction that effective viewing distances are defined such that the effective display width having a 3D crosstalk under 300% must be over 50% of the display screen size, then the effective viewing distances which are valid are in the range from 2.3 m to 2.7 m, and the effective viewing range is 0.4 m which is 16% of the optimal viewing distance of ze = 2.5 m.

4. Conclusion

In this study, a 3D simulation model calculating the intensity distribution for the entire display screen of an autostereoscopic display at a given eye position was developed. A parallax barrier array was used as an optical model and the reverse ray tracing of light from the observer’s eye to the subpixels through the slits of the barrier were performed based on Snell’s law of refraction using the inclination and azimuthal angles of the light rays for the 3D simulation. By investigating the optical behaviors of the displayed image of the nine-view autostereoscopic display for various viewing distances under an actual design condition, we found the inhomogeneous light leakages from the unwanted views and predicted the segmented images which were comprised of multiple images from other views on the entire display screen. We also found that the displayed images are shown to be mixed and segmented and that the order of the segmented image is in forward order at viewing distances located far from the optimal viewing distance and in reverse order at viewing distances between the display and the optimal viewing distance.

The FWHMs of the intensity distributions on the display screen rapidly increased in size from 30% to 100% of the width of the display screen for the viewing distances from 2.1 m to 2.4 m and slowly decreased in size again from 100% to 35% of the width of the display screen for the viewing distances from 2.5 m to 2.9 m. For the optimal viewing distance, the 3D crosstalk was calculated to be 100% for the entire display screen in the horizontal direction, as expected in nine-view autostereoscopic displays, but increased abruptly when the observer departs from the optimal viewing distance. If we make a restriction that effective viewing distances are defined such that the effective display width having a 3D crosstalk under 300% must be over 50% of the display screen size, then the effective viewing distances which are valid are in the range from 2.3 m to 2.7 m, and the effective viewing range is obtained to be 0.4 m, which is 16% of the optimal viewing distance, where ze = 2.5 m. From the results, it is revealed that our simulation model has good potentiality for predicting the displayed image on the entire display screen of autostereoscopic displays for various positions of the observer’s eye with fast calculation speed.

Acknowledgment

The authors thank their colleagues and Dr. Kyoung-Moon Lim in LG Display Co., Ltd. for useful discussions and special support in developing this work.

References and links

1. H.-K. Hong, S.-M. Jung, B.-J. Lee, H.-J. Im, and H.-H. Shin, “Autostereoscopic 2D/3D switching display using electric-field-driven LC lens(ELC lens),” SID Symp. Digest Tech. Papers 39, 348−351 (2008). [CrossRef]  

2. H.-J. Im, S.-M. Jung, B.-J. Lee, H.-K. Hong, and H.-H. Shin, “Mobile 3D displays based on a LTPS 2.4” VGA LCD panel attached with lenticular lens sheets,” SID Symp. Digest Tech. Papers 39, 256−259 (2008).

3. M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008). [CrossRef]  

4. M.-C. Park, H.-D. Lee, and J.-Y. Son, “Interactive 3D simulator for autostereoscopic display systems,” in Proceedings of International Display Workshops (2011), pp. 1849−1851.

5. S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013). [CrossRef]  

6. S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013). [CrossRef]  

7. S.-M. Jung and I.-B. Kang, “Three-dimensional modeling of light rays on the surface of a slanted lenticular array for autostereoscopic displays,” Appl. Opt. 52(23), 5591–5599 (2013). [CrossRef]   [PubMed]  

8. S.-M. Jung and I.-B. Kang, “Numerical simulation of the optical characteristics of autostereoscopic displays that have an aspherical lens array with a slanted angle,” Appl. Opt. 53(5), 868–877 (2014). [CrossRef]   [PubMed]  

9. C. Berkel, “Image preparation for 3D-LCD,” Proc. SPIE 3639, 84–91 (1999). [CrossRef]  

10. G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000). [CrossRef]  

11. E. Hecht, Optics (Addison-Wesley Publishing Co., 1987).

12. A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

References

  • View by:

  1. H.-K. Hong, S.-M. Jung, B.-J. Lee, H.-J. Im, and H.-H. Shin, “Autostereoscopic 2D/3D switching display using electric-field-driven LC lens(ELC lens),” SID Symp. Digest Tech. Papers 39, 348−351 (2008).
    [Crossref]
  2. H.-J. Im, S.-M. Jung, B.-J. Lee, H.-K. Hong, and H.-H. Shin, “Mobile 3D displays based on a LTPS 2.4” VGA LCD panel attached with lenticular lens sheets,” SID Symp. Digest Tech. Papers 39, 256−259 (2008).
  3. M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008).
    [Crossref]
  4. M.-C. Park, H.-D. Lee, and J.-Y. Son, “Interactive 3D simulator for autostereoscopic display systems,” in Proceedings of International Display Workshops (2011), pp. 1849−1851.
  5. S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
    [Crossref]
  6. S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
    [Crossref]
  7. S.-M. Jung and I.-B. Kang, “Three-dimensional modeling of light rays on the surface of a slanted lenticular array for autostereoscopic displays,” Appl. Opt. 52(23), 5591–5599 (2013).
    [Crossref] [PubMed]
  8. S.-M. Jung and I.-B. Kang, “Numerical simulation of the optical characteristics of autostereoscopic displays that have an aspherical lens array with a slanted angle,” Appl. Opt. 53(5), 868–877 (2014).
    [Crossref] [PubMed]
  9. C. Berkel, “Image preparation for 3D-LCD,” Proc. SPIE 3639, 84–91 (1999).
    [Crossref]
  10. G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
    [Crossref]
  11. E. Hecht, Optics (Addison-Wesley Publishing Co., 1987).
  12. A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

2014 (1)

2013 (3)

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
[Crossref]

S.-M. Jung and I.-B. Kang, “Three-dimensional modeling of light rays on the surface of a slanted lenticular array for autostereoscopic displays,” Appl. Opt. 52(23), 5591–5599 (2013).
[Crossref] [PubMed]

2008 (1)

M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008).
[Crossref]

2000 (1)

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

1999 (1)

C. Berkel, “Image preparation for 3D-LCD,” Proc. SPIE 3639, 84–91 (1999).
[Crossref]

Berkel, C.

C. Berkel, “Image preparation for 3D-LCD,” Proc. SPIE 3639, 84–91 (1999).
[Crossref]

Boev, A.

A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

Egiazarian, K.

A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

Ezra, D.

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

Gotchev, A.

A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

Harrold, J.

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

Jacobs, A. M. S.

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

Jang, J.-H.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Järvenpää, T.

M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008).
[Crossref]

Jung, S.-M.

S.-M. Jung and I.-B. Kang, “Numerical simulation of the optical characteristics of autostereoscopic displays that have an aspherical lens array with a slanted angle,” Appl. Opt. 53(5), 868–877 (2014).
[Crossref] [PubMed]

S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
[Crossref]

S.-M. Jung and I.-B. Kang, “Three-dimensional modeling of light rays on the surface of a slanted lenticular array for autostereoscopic displays,” Appl. Opt. 52(23), 5591–5599 (2013).
[Crossref] [PubMed]

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Kang, H.-Y.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Kang, I.-B.

Kang, J.-N.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Lee, K.-J.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Lee, S.-C.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
[Crossref]

Lim, K.-M.

S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
[Crossref]

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Moseley, R. R.

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

Salmimaa, M.

M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008).
[Crossref]

Woodgate, G. J.

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

Yeo, S.-D.

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Appl. Opt. (2)

Curr. Appl. Phys. (1)

S.-M. Jung, S.-C. Lee, and K.-M. Lim, “Two-dimensional modeling of optical transmission on the surface of a lenticular array for autostereoscopic displays,” Curr. Appl. Phys. 13(7), 1339–1343 (2013).
[Crossref]

J. Soc. Inf. Disp. (1)

M. Salmimaa and T. Järvenpää, “3-D crosstalk and luminance uniformity from angular luminance profiles of multiview autostereoscopic 3-D displays,” J. Soc. Inf. Disp. 16(10), 1033–1040 (2008).
[Crossref]

Proc. SPIE (3)

C. Berkel, “Image preparation for 3D-LCD,” Proc. SPIE 3639, 84–91 (1999).
[Crossref]

G. J. Woodgate, J. Harrold, A. M. S. Jacobs, R. R. Moseley, and D. Ezra, “Flat panel autostereoscopic displays-characterisation and enhancement,” Proc. SPIE 3957, 153–164 (2000).
[Crossref]

S.-M. Jung, J.-H. Jang, H.-Y. Kang, K.-J. Lee, J.-N. Kang, S.-C. Lee, K.-M. Lim, and S.-D. Yeo, “Optical modeling of a lenticular array for autostereoscopic displays,” Proc. SPIE 8648, 864805 (2013).
[Crossref]

Other (5)

E. Hecht, Optics (Addison-Wesley Publishing Co., 1987).

A. Boev, A. Gotchev, and K. Egiazarian, “Crosstalk measurement methodology for auto-stereoscopic screen,” in Proceedings of IEEE 3D TV Conference (IEEE, 2007), pp. 1−4.

M.-C. Park, H.-D. Lee, and J.-Y. Son, “Interactive 3D simulator for autostereoscopic display systems,” in Proceedings of International Display Workshops (2011), pp. 1849−1851.

H.-K. Hong, S.-M. Jung, B.-J. Lee, H.-J. Im, and H.-H. Shin, “Autostereoscopic 2D/3D switching display using electric-field-driven LC lens(ELC lens),” SID Symp. Digest Tech. Papers 39, 348−351 (2008).
[Crossref]

H.-J. Im, S.-M. Jung, B.-J. Lee, H.-K. Hong, and H.-H. Shin, “Mobile 3D displays based on a LTPS 2.4” VGA LCD panel attached with lenticular lens sheets,” SID Symp. Digest Tech. Papers 39, 256−259 (2008).

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Figures (8)

Fig. 1
Fig. 1 Optical configuration of our 3D simulation model calculating the intensity distribution over the entire screen of the autostereoscopic display with the parallax barrier array. The coordinate systems, planes of interface, parallax barrier and the subpixel array are described. The wave vectors of the incident and the transmitted light rays are also depicted in the figure.
Fig. 2
Fig. 2 Illustrations of the calculation domains for (a) the parallax barrier array, (b) the subpixel array and (c) the display screen of the autostereoscopic display. The starting point (S) and crossing point (C) of each light ray are introduced. The slanted angle ϕs and horizontal barrier pitch Ph of the barrier array, and the subpixel sizes Px and Py in the x- and y-directions are defined. The rectangular grids on the display screen are also illustrated in the figure.
Fig. 3
Fig. 3 Simulation procedure for obtaining the intensity distribution over the entire display screen. After configuring all the numerical and physical parameters, reverse tracing for each light ray from the starting point to the observer’s eye is performed. All the intensities of the individual light rays are collected at each grid point corresponding to their position in relation to the crossing points on the display screen.
Fig. 4
Fig. 4 The design condition of nine-view autostereoscopic displays used in our simulation. The view map arrangement of subpixels assigned to all the view numbers is illustrated. We used the subpixel group of nine-view autostereoscopic displays having 4.5 subpixels and 2 lines in the horizontal and vertical directions. The intended view map angle of our configuration was designed to be φs = tan−1(1/6) = 9.462° in our configuration of the simulation.
Fig. 5
Fig. 5 Calculated intensity distribution on the display screen observed at the optimal viewing distance of ze = 2.5 m under various test images for which subpixels corresponding to a given view are white and the others are black. The observer’s eyes are located at the center front of the display, i. e., xe = 0 and ye = 0, where view 5 among the nine views can be observed over the entire display screen. Even though the observer’s eyes are located to see the full white image of view 5, the neighboring images for view numbers 4 and 6 are also observed for almost the whole display screen, and views for 3 and 7 are also found around the corners of the display screen.
Fig. 6
Fig. 6 Calculated intensity distribution over the display screen observed at various viewing distances from z = 0.5 m to 4.5 m with a step size of 0.5 m for a test image for which the subpixels correspond to view 5. The white regions in accordance with the viewing distance are found where the image for view 5 is displayed on the display screen.
Fig. 7
Fig. 7 Cross sectional intensity distribution on the display screen observed at various viewing distances from z = 2.1 m to 2.9 m with a step size of 0.1 m for various test images of the various views. The FWHM of each intensity distribution increases gradually for viewing distances between 2.1 m and 2.5 m and decreases again for viewing distances greater than 2.5 m.
Fig. 8
Fig. 8 Cross sectional 3D crosstalk distributions in the horizontal direction of the display screen under various viewing distances. The 3D crosstalk value around the center of the display screen is 100% for all the viewing distances and increases as the position on the display screen becomes far from the center of the display screen at viewing distances other than the optimal viewing distance.

Tables (1)

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Table 1 List of Parameters and Values Used in the Simulation

Equations (20)

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k i = 2π λ n i (sin θ i cos φ i x+sin θ i sin φ i y+cos θ i z),
k t = 2π λ n t (sin θ t cos φ t x+sin θ t sin φ t y+cos θ t z).
θ t = tan 1 [ ( x e x c ) 2 + ( y e y c ) 2 z e z c ],
φ t = tan 1 [ ( y e y c ) ( x e x c ) ].
θ i = sin 1 [( n t / n i ) sin θ t ],
φ i = φ t .
x s =tan θ i cos φ i ( z c z s )+ x c ,
y s =tan θ i sin φ i ( z c z s )+ y c ,
z s =g.
x=tan φ s y+ P h l+ x b .
x c =tan φ s y c + P h l+ x b ,
y c =Δym,
z c =0.
x s / P x + N x /21<i x s / P x + N x /2,
y s / P y + N y /21j< y s / P y + N y /2.
v ij = N t Mod[i+1(j+1)( P y / P x )tan ϕ s + x os , N h ]( N t N h ).
T = n t cos θ t n i cos θ i [ 2 n i cos θ i n i cos θ i + n t cos θ t ] 2 ,
T ll = n t cos θ t n i cos θ i [ 2 n i cos θ i n i cos θ t + n t cos θ i ] 2 .
I t ( θ t , φ t )= 1 2 ( T + T ll ) I i ( θ i , φ i ).
X i (x,y)= 1 L i (x,y) [ j=1 j= N t L j (x,y) L i (x,y) ]×100%

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