1. Introduction

Presently, liquid crystal display (LCD) and organic light-emitting diode (OLED) display are two dominating technologies for display applications [1]. Moreover, mini-LED and micro-LED displays are attracting extensive attentions for their optical and structural advantages [2]. LCDs are non-emissive, while OLEDs, mini-LED and micro-LED displays are emissive. The emissive technologies have their inherent advantage of true black state, while the non-emissive technology does not. In order to compete with other technologies, LCD with High Dynamic Range (HDR) has emerged as an attractive technology for its excellent image qualities in recent years [3]. With high contrast ratio in the HDR display, detailed images can be revealed well in both high and low brightness regions simultaneously [4]. Besides, HDR demands wider color gamut and greater bit depth to provide more frame details, all of which contribute to display the world as authentic as what we see with our own eyes. Along with HDR technology popularizing in the entertainment industry, there are a series of applications of HDR technology in large-scale displays. For such HDR applications, one way is to develop algorithms that merge low dynamic range images into HDR image [5]. The other way is to develop HDR hardware such as HDR backlights and HDR panels [6,7]. Dual-panel LCD TV module has been developed to improve picture quality [8,9]. It works by stacking two LCD panels; one for modulating light and the other for controlling color. The contrast could be drastically increased with the upper panel acting as normal display panel and the under panel realizing the HDR function. Both panels have black matrix (BM) that blocks the wires on the panel. The regular layouts of BM lead to the moiré patterns.

The color moirés could appear in displays when the sub-pixels are blocked regularly and the pattern of moiré is large enough to distinguish the colors. Since they deteriorate the image quality of the displays, they are highly undesired interference phenomena that have to be minimized or eliminated in the displays [10]. In this regard, some methods have been developed to analyze and describe the color fringe patterns, trying to alleviate the moiré pattern [11]. Based on the conventional beat frequency formula that works well when the periods of the two line patterns have a small difference, some researchers have developed a new formula to describe the period of moiré for 3-D displays, in which case the line pattern is usually more than twice the pixel pattern [12]. But the result calculated by the developed formula is inconsistent with the simulation for a case of two line patterns of 0.372 and 0.820 mm. The period of color moiré is calculated by Eq. (1) in [12] and the result is 6.4 mm. While the simulation indicates that the period value is about 4 mm. It implies that the developed formula may not be correct for some cases of two line patterns. Furthermore, the developed formula in [12] can’t be applied for two line patterns with a rotation angle. Therefore, an accurate formula of the moiré pattern is still urgently needed. The simulation methods of color moirés have also been developed [13,14]. The simulation can be used as a powerful tool to verify the accuracy of the formula.

In this paper, a new mathematical formula is derived to calculate the fringe periods of the color moirés appearing on the dual-panel display. And the applicable situation for the formula is demonstrated. The calculation results have been compared with the experiment and simulation results of moiré patterns in dual-panel displays. The consistency of the results has corroborated the validity of the formula.

2. Derivation of moiré fringe period in the dual-panel display

The upper panel and the under panel are separated by a glass plate and an air plate in the dual-panel display, as shown in Fig. 1. The glass plate has a thickness of t and the air plate has a thickness of w. As the blocked number (or portion) of the sub-pixel(s) of upper panel will be increased as the viewing angle increases for a given viewing distance, the color moiré fringes could appear on the upper panel. The viewing geometry of the two overlapped regular patterns can be described in Fig. 1. The geometrical optics approach is used to analyze the color moiré fringes and the diffraction of the panels is neglected. Because the diffraction has been averaged out for a Lambertian light source, which is similar to the backlight [15]. In Fig. 1, the center points of first and second BM line counting from the center of the under panel are specified as ${A_1}$ and ${A_2}$, respectively. ${D_1}$ and ${D_2}$ are the matching points at the vertical direction on the upper panel. BM will block the light propagation. In order to describe the blocking effect conveniently, it is assumed that BM is transparent. In this way, we can easily find out the location of the black matrices’ shadow on the upper panel. As the ray propagate from ${A_1}$ and ${A_2}$, ${B_1}$ and ${B_2}$ are the matching points below the glass plate, ${C_1}$ and ${C_2}$ are the matching points above the glass plate. The ray propagation from ${B_1}$ to ${C_1}$ and ${B_2}$ to ${C_2}$ has been refracted by the glass plate. $\Delta {d_1}$ represents the distance between ${C_1}$ and ${D_1}$. $\Delta {d_2}$ represents the distance between ${C_2}$ and ${D_2}$_. The rays finally reach the eye specified as point E with viewing angles of $G{\theta _1}$ and $G{\theta _2}$. If there is no glass plate, the rays will propagate through ${C_1}^{\prime}$ and ${C_2}^{\prime}$ and reach the eye specified as point E with viewing angles of ${\theta _1}$ and ${\theta _2}$.

 

Fig. 1. A geometry of forming color moiré fringes.

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$\Delta {d_1}$ and $\Delta {d_2}$ are different due to the different blocking effect of ${C_1}$ and ${C_2}$. The difference of the blocking effect of ${C_1}$ and ${C_2}$ is corresponding to the relative movement of ${C_1}$ and ${C_2}$, which can be expressed as,

In Fig. 1, other parameters such as the viewing distance from the panel, the period of under panel in horizontal direction and the refractive index of the glass material are specified as h, ${P_1}$, and n, respectively. Furthermore, the distances in the horizontal direction between points ${A_1}$ and ${A_2}$, ${C_1}$ and $O$, ${C_1}$ and ${D_1}$, ${C_1}$ and ${B_1}$, ${B_1}$ and O, ${D_1}$ and O, ${C_2}$ and O, ${C_2}$ and ${D_2}$, ${C_2}$ and ${B_2}$, ${B_2}$ and O, ${D_2}$ and O, ${B_1}$ and ${D_1}$, ${B_2}$ and ${D_2}$, ${D_1}$ and ${D_2}$, are specified as $d({A_1}{A_2})$, $d({C_1}O)$, $d({C_1}{D_1})$, $d({C_1}{B_1})$, $d({B_1}O)$, $d({D_1}O)$, $d({C_2}O)$, $d({C_2}{D_2})$, $d({C_2}{B_2})$, $d({B_2}O)$, $d({D_2}O)$, $d({B_1}{D_1})$, $d({B_2}{D_2})$ and $d({D_1}{D_2})$. The following equations can be obtained from the geometry relationship in Fig. 1. Equations (3)–(13) describe the distances for $d({C_1}O)$, $d({C_1}{B_1})$, $d({B_1}{D_1})$, $d({D_1}O)$, $d({C_1}{D_1})$, $d({C_2}O)$, $d({C_2}{B_2})$, $d({B_2}{D_2})$, $d({D_2}O)$, $d({C_2}{D_2})$ and $d({D_1}{D_2})$, respectively.

As $\Delta {d_1} = d({C_1}{D_1})$, ${d_1} = d({D_1}O)$, $\Delta {d_2} = d({C_2}{D_2})$, ${d_2} = d({D_2}O)$ and ${P_1} = d({D_1}{D_2})$ can be easily obtained in Fig. 1, the angel of $G{\theta _1}$ and $G{\theta _2}$ can be derived from the following equations,

It is clear shown that ${\Delta _i}$ is not constant as the value of $[\tan (G{\theta _{i + 1}}) - \tan (G{\theta _i})]$ changes. If $\Delta $ reaches ${P_2}$ when $i = j$, then a full color moiré can be obtained. As ${\Delta _i}$ is not constant, the period P of the color moiré is not constant and the formula for P will be very complex.

In order to simplify the formula for P, the propagations of the ray path ${A_1}{B_1}{C_1}E$ with glass plate and the ray path ${A_1}{C_1}^{\prime}E$ without glass plate are compared. The ray propagates from ${A_1}$ and ${A_2}$ through ${C_1}^{\prime}$ and ${C_2}^{\prime}$ and then reaches the eye without the glass plate, and it has also been shown in Fig. 1. The equations can be obtained for ray path ${A_1}{C_1}^{\prime}E$ just the same way for ray path ${A_1}{B_1}{C_1}E$. Eq. (18) and Eq. (19) describe the distance for $d({C_1}^{\prime}O)$ and $d({C_2}^{\prime}O)$. Eq. (20) and Eq. (21) describe the calculated angle for ${\theta _1}$ and ${\theta _2}$. Eq. (22) describes the relative movement of ${C_1}^{\prime}$ and ${C_2}^{\prime}$.

Equation (22) can be rewritten as Eq. (23), as ${P_1} = d({D_1}{D_2}) = {d_2} - {d_1}$ and $t + w$ is replaced as $\Delta h$. ${\Delta _i}^{\prime}$ can be calculated in the same way which can be express as Eq. (24). It is clear shown that ${\Delta _i}^{\prime}$ is constant. Considering the process for the formation of color moiré, the period of the color moiré can be obtained by dividing ${P_2}$ by ${\Delta _i}^{\prime}$ and it is expressed as Eq. (25).

Equation (25) is the derived formula for the period of the color moiré in dual-panel display in the situation of ${P_1} = k\ast {P_2}$. For the situation of ${P_1} \ne k\ast {P_2}$, Eq. (25) will be some kind of different. The relative movement of ${C_i}^{\prime}$ and ${C_{i + 1}}^{\prime}$ can be expressed as $[{\Delta _i}^{\prime} - ({P_1} - k\ast {P_2})]$, where $k = [{P_1}/{P_2}]$ represents the integer value produced by dividing ${P_2}$ into ${P_1}$. The value of k will be discussed carefully later. Equation (25) should be rewritten as Eq. (26) for the situation of ${P_1} \ne k\ast {P_2}$.

The width of the BM is discussed for Eq. (26). In Fig. 1, when the left and right edge points of the i-th line from the center of the panel are specified as F and G, respectively, F and G will be matched to the points $F^{\prime}$ and $G^{\prime}$ on the panel, respectively. The distance between points F and G is the length of BM, which is ${P_{FG}}$. The distance ${P_{F^{\prime}G^{\prime}}}$ between points $F^{\prime}$ and $G^{\prime}$ determines the number of sub-pixels to be blocked by the i-th line. From the geometry in Fig. 1, the relation between ${P_{FG}}$ and ${P_{F^{\prime}G^{\prime}}}$ can be expressed as Eq. (27).

There may be an angle between the dual panels. In the above discussion, the angle between two structures has not been considered. In a simple way, the angle is considered for the display with $\Delta h = 0$. For this situation, the formula for P can be derived from geometry that has been largely reported [16–18]. The formulas for $P$ and $\beta $ are Eq. (28), Eq. (29) and Eq. (30).

In the above formulas, $\alpha $ is the angle between two linear structures and $\beta $ is the angle of the color moiré pattern. It should be noticed that there is just a little difference between Eq. (26) and Eq. (28) when $\alpha = {0^ \circ }$. This implies that formulas can be written in the same form in the case of $\alpha \ne {0^ \circ }$. If the $\Delta h$ is considered, the formulas for P and $\beta $ should be written as Eq. (31), Eq. (32), and Eq. (33).

The above equations are the final formulas that can be used to calculate the moiré pattern in dual-panel display with air plate at normal viewer distance. In the equations, the value of k should be discussed particularly. Let $P(k) = P(k + 1)$ and Eq. (34) can be derived.

Table 1. The rules for the value of k.^a

View Table

Therefore, the formula for P is very simple for dual-panel display with air plate. In order to find out the condition for using the equation for glass plate, the difference is compared for the relative movement $[{\Delta _i}^{\prime} - ({P_1} - k\ast {P_2})]$ between ${C_i}^{\prime}$ and ${C_{i + 1}}^{\prime}$ for air plate and the relative movement $[{\Delta _i} - ({P_1} - k\ast {P_2})]$ between ${C_i}$ and ${C_{i + 1}}$ for glass plate. From the above discussion, the difference of $[{\Delta _i}^{\prime} - ({P_1} - k\ast {P_2})]$ and $[{\Delta _i} - ({P_1} - k\ast {P_2})]$ is represented as $\Delta \Delta $ which can be expressed as Eq. (35). Eq. (36) describes the relatively proportion of $\Delta \Delta $ in the full movement, which can be expressed as $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$.

$\Delta \Delta $ and $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ are numerically calculated for a specific dual-panel display in Fig. 2(a) and Fig. 2(b). The parameters are ${P_1} = 0.744mm$, ${P_2} = 0.372mm$, $s = 100mm$, $t = 0.64mm$, $w = 0.02mm$ and $n = 1.5$. Fig. 2(a) shows a graph of $\Delta \Delta $ versus the viewing distance when ${P_1}$ is 0.744, 0.748, 0.758 and 0.768 mm, respectively. It indicates that the value of $\Delta \Delta $ is less than 1 µm, which is much smaller than a pixel. $\Delta \Delta $ decreases as h increases and they almost have the same value, regardless of ${P_1}$. Figure 2(b) shows a graph of $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ versus the viewing distance when ${P_1}$ is 0.744, 0.748, 0.758 and 0.768 mm, respectively. It indicates that the relatively proportion of $\Delta \Delta $ in the full movement is quite different depending on ${P_1}$. $\Delta \Delta $ is quite small compared to the full movement ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$ when ${P_1}$ is far away from $k\ast {P_2}$. The period of color moiré relies on ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$. ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$ is almost ${{\Delta }_i}^{\prime}$ when ${P_1}$ is nearby $k\ast {P_2}$ and ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$ is almost $({{P_1} - k\ast {P_2}} )$ when ${P_1}$ is far away from $k\ast {P_2}$. So it can be deduced that the difference of the periods of color moiré for air plate and glass plate is slightly large when ${P_1}$ is nearby $k\ast {P_2}$, as $\Delta \Delta $ is quite large compared to the full movement. For this situation, the period of the color moiré for the air plate will be smaller than the glass plate, as $\Delta \Delta $ is positive and ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$ is slightly larger than $[{\Delta _i} - ({P_1} - k\ast {P_2})]$. The difference of the periods of color moiré for air plate and glass plate is quite small when ${P_1}$ is far away from $k\ast {P_2}$, as $\Delta \Delta $ is quite small compared to the full movement. For this situation, the period of the color moiré for the air plate will be similar to the glass plate, as ${{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )$ is almost equal to $[{\Delta _i} - ({P_1} - k\ast {P_2})]$. While keep the difference of glass plate and air plate in mind, the glass plate can be considered as air plate for convenient analyzing the color moiré. The equations for ray path ${A_1}{C_1}^{\prime}E$ can replace the equations for ray path ${A_1}{B_1}{C_1}E$.

 

Fig. 2. (a) $\Delta \Delta $ and (b) $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ changes with different viewing distance when ${P_1}$ is 0.744, 0.748, 0.758 and 0.768 mm, respectively.

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Based on the equations obtained for the color moiré pattern in dual-panel display, some phenomena can be anticipated easily both for air plate and glass plate. Firstly, when $\alpha = {0^ \circ }$ and ${P_1}$ is far away from $k\ast {P_2}$, the value of P will increase as the value of h decreases. It implies that the pattern seems to expand in horizontal direction as the observer gets closer to the display system. Secondly, the value of P will decrease as the value of $\alpha $ increases. It indicates that the pattern seems to narrow as the angle between dual panels increases.

The period of color moiré that is related to the under panel period has been calculated for the dual-panel display with the derived equations. The results are shown in Fig. 3. It indicates that there are two kinds of P, P on the peak and P in the valley, specified as ${P_P}$ and ${P_V}$, respectively. ${P_P}$ is infinite at some special values of ${P_1}$. The values can be derived when the denominator of Eq. (31) is zero. Eq. (37) is the expression of the denominator when it is zero.

 

Fig. 3. The pitch of color moiré as a function of the under panel period ${P_1}$ when $\alpha = {0^ \circ }$.

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The phenomena of color moiré for ${P_P}$ and ${P_V}$ are compared in Fig. 4 as ${P_1}$ is 0.744 mm for ${P_P}$ and 0.768 mm for ${P_V}$. The value of P increases lineally versus the viewing distance in Fig. 4(a) when ${P_1}$ is 0.744 mm. Figure 4(b) indicates that the value of P decreases very slightly versus the viewer distance when ${P_1}$ is 0.768 mm. The phenomena of color moiré for ${P_P}$ and ${P_V}$ are quite different and can be verified through the simulation of color moiré.

 

Fig. 4. The pitch of color moiré as a function of the under cell pitch ${P_1}$ when $\alpha = {0^ \circ }$.

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3. Experiment and simulation of color moiré

Some dual-panel LCD TV samples with different under panel periods are used in the experiment. For the experiment, the pixel size of the test sample is same with a 65-inch full-size TV, and the size of the sample is only 10 inch. The pixel size of the upper panel is 0.372 mm (horizontal, H) by 0.372 mm (horizontal, V). The pixel size of the under panel of sample 1 is 0.768 mm (horizontal, H) by 0.744 mm (horizontal, V), and the rotational angel $\alpha $ is 0°. The pixel size of the under panel of sample 2 is 0.744 mm (horizontal, H) by 0.744 mm (horizontal, V), and the rotational angel $\alpha $ is 1.8°. The gap between the dual panels is 0.66 mm. A camera is used to capture the color moiré pattern at the specific distance. Fig. 5(a) and Fig. 5(c) show the experimental results of a color moiré pattern when the panel displays a white image. We simulated the color moiré with Lighttools software and inserted the same system parameters in the simulation as the experimental setup. Figures 5(b) and 5(d) show the simulated results of the color moiré patterns. The observed color moiré patterns are compared with the simulation images. The experimental results are just like as the discussion in the period derivation section: with the blocking effect of BM, the color moiré in Fig. 5 appear three colors: cyan, purple and yellow, when the sub pixel of R, G and B is blocked by BM of under panel. The periods of simulated color moiré in Fig. 5(b) and Fig. 5(d) are 13.3 and 11.2 mm, respectively. They meet well with the experimental results in Fig. 5(a) and Fig. 5(c), which is 13.2 and 11.2 mm. The experimental results agree well with the simulation results, which verifies the validity of the proposed moiré pattern simulation. If the system parameters are known, the color moiré pattern according to the slant angle can be anticipated easily by the proposed simulation. This is a distinct advantage of the proposed simulation method, which enables us to visualize the moiré pattern according to the slant angle.

 

Fig. 5. Comparisons of experimentally and simulated obtained color moiré fringes at viewing distance of 200 mm, (a) and (b) under cell pitch is 768 µm at the horizontal direction and the rotated angel $\alpha $ is 0°. (c) and (d) under cell pitch is 744 µm at the horizontal direction and the rotated angel $\alpha $ is 1.8°. The length unit of scale bar in (b) and (d) is millimeter (mm).

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In the discussion of the period derivation section, it is demonstrated that the period of the color moiré for the air plate will be similar to the glass plate when P₁ is far away from $k\ast {\textrm{P}_2}$. In order to illustrate this phenomenon at different viewing distances, the simulation has been conducted at three viewer distance of 35, 200 and 500 mm for two situations of glass gap and air gap with ${P_1}$ is 0.768 mm and ${P_2}$ is 0.372 mm. Fig. 6 shows the simulated results. There is a big mismatch between Fig. 6(a) and Fig. 6(d) at the viewing distance of 35 mm. In Fig. 6(a), the moiré fringe appears curve obviously, while it is straight in Fig. 6(d). It has been discussed in the in the period derivation section that the numerically calculated $\Delta \Delta $ and $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ increase as h decrease. The increased $\Delta \Delta $ and $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ leads to the big mismatch between Figs. 6(a) and 6(d) at the viewing distance of 35 mm. The mismatch become small at the viewing distance of 200 mm. The moiré fringes are both straight in Fig. 6(b) and Fig. 6(e). There is only a little difference of the moiré pitch. The mismatch is too small to observe at the viewer distance of 500 mm and difference of the moiré periods is negligible in Fig. 6(c) and Fig. 6(f). The simulated results demonstrate that the glass plate can be considered as air plate for the viewer distance $h > 500mm$ as illustrated in the discussion of the period derivation section. Also equations for ray path ${A_1}{C_1}^{\prime}E$ are approximate equations for ray path ${A_1}{B_1}{C_1}E$ when $h > 500mm$ and ${P_1}$ is far away from $k\ast {P_2}$. Employ the equation for the three viewing distances of 35, 200 and 500 mm, the calculated periods are 28.7, 13.3 and 12.4 mm, respectively. The calculated results are accordant to the simulated results in Fig. 6(d), Fig. 6(e) and Fig. 6(f).

 

Fig. 6. Comparisons of simulated color moiré fringes for two situations at viewer distance of 35 mm, 200 mm and 500 mm. (a), (b) and (c) the glass gap. (d), (e) and (f) the air gap. The length unit of scale bar in the figures is 20 millimeter (mm).

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The phenomenon of the color moiré shown in Fig. 7 are demonstrated for the glass plate and the air plate with different angles, for which ${P_1}$ is 0.744 mm and ${P_2}$ is 0.372 mm. For the glass plate, the periods of the simulated color moiré are 216, 153 and 21 mm, and the angles $\gamma $ are 180°, 135° and 96° when $\alpha $ is 0°, 0.1° and 1°. The angle $\gamma $ is illustrated in Fig. 8. For the air plate, the periods of the simulated color moiré are 141, 119 and 21 mm, and the angles $\gamma $ are 180°, 145° and 98° when $\alpha $ is 0°, 0.1° and 1°. The difference for both periods and angles of the color moiré between the glass and air plate decrease dramatically when $\alpha $ increases. There is no obvious difference when $\alpha $ is 1°. It implies that the equations for ray path in air plate can be used for ray path in glass plate when α is slightly larger than 0°, like 1°. By using the equations, the calculation for the color moiré of the glass plate will be very convenient. The period and angle of color moiré along with the angle α are demonstrated in Fig. 8. The calculated periods are 141, 117 and 21 mm and the angles are 180°, 146.6° and 99.1° when $\alpha $ is 0°, 0.1° and 1°. The calculated angle results are $\beta $ and the simulated angle results are $\gamma $, as illustrated in Fig. 8. Their relationship can be expressed as: $\gamma = \beta - \alpha $. Employing the equation to the calculated results, the calculated ${\gamma }$ is 180°, 146.5° and 98.1°. The results are accordant to the simulated results in Fig. 7(d), Fig. 7(e), and Fig. 7(f). It should be noticed that the formula derived is used to calculate the period of color moiré between two line patterns. It is not suitable to directly employ the formula to calculate the period of color moiré among four line patterns when the slanting angle of two panels is large. In order to avoid the situation, the simulated slanting angle is small enough to make sure that the formula is effective. The period of color moiré among four line patterns can be calculated one by one, when the slanting angle of two panels is large. To do this, the formula for two line patterns is the base. The more complicated calculation will be our future research.

 

Fig. 7. Comparisons of simulated color moiré fringes for two situations with $\alpha $ is 0°, 0.1° and 1° at viewer distance of 250 mm. (a), (b) and (c) the glass gap. (d), (e) and (f) the air gap.

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Fig. 8. The schematic diagram of color moiré pattern for dual-panel display when the rotational angle α is slightly larger than 0°.

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In order to demonstrate the difference of color moiré for glass and air precisely, the difference of $\Delta \Delta $ and $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ is calculated when $\alpha $ is 1°. The calculation method is similar to the method in the discussion of the period derivation section, except that ${P_1}$ will be ${P_1}^{\prime}$ and ${P_2}$ will be ${P_2}^{\prime}$, as illustrated in Fig. 8. The expression for ${P_1}^{\prime}$ is ${P_1}/\sin (\beta - {90^ \circ })$ and the expression for ${P_2}^{\prime}$ is ${P_2}/\sin (\beta - \alpha - {90^ \circ })$. When $\alpha $ is 1°, $\beta $ can be calculated which is 99.1°, ${P_1}^{\prime}$ is 4.70 mm and ${P_2}^{\prime}$ is 2.64 mm. The calculated $\Delta \Delta $ is 0.005 mm and $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ is 0.84%. $\frac{{{\Delta \Delta }}}{{{{\Delta }_i}^{\prime} - ({{P_1} - k\ast {P_2}} )}}$ represents the relatively proportion of $\Delta \Delta $ in the full movement. $\Delta \Delta $ is quite small compared to the full movement when $\alpha $ is 1°. It implies that the difference of color moiré for glass and air plate is negligible.

4. Conclusion

A mathematical formula to calculate the fringe periods of the color moirés appearing at the dual panel displays is derived. The difference between glass plate and air plate is compared carefully both on calculation and simulation. The results demonstrate that the derived formula can be applied to the glass plate only except when ${P_1}$ is nearby $k\ast {P_2}$ and $\alpha $ is around 0° at the same time. The developed calculation method has a wide range of applications in calculating the periods of moiré patterns, such as the moiré patterns formed by two structures of the panel and the prism in backlight.