A precise and fast computational method for the simulation and analysis of moiré patterns is proposed. This new algorithm is based on convolution with superposition of the intensity profile which is transmitted from the optical layers and the point spread function. The computational time is shown to be much faster than that of the ray-tracing algorithm because the new algorithm does not involve a massive calculation. Also, information on the moiré pitch can be extracted directly from the sampling data of the moiré patterns.
© 2014 Optical Society of America
A display system comprises various optical film components with periodic structure, which are intended to enhance various aspects of display quality, such as brightness, color performance, and viewing angle of display [1, 2]. These structures of the various display systems have pitch ranges from tens to hundreds micrometers, and they are designed to minimize product thickness with the resolution of the human eye in mind [1–4]. However, moiré effect stemming from the superposition of the transmitted intensity profile from each periodic component may degrade the image quality in the display system if the pitch of the moiré pattern is below the limit of the eye resolution and the amplitude is sufficiently high [5, 6].
In principle, moiré patterns are inevitable in display systems with multiple periodic optical layers, but their pitch and amplitude can be reduced by tuning the pitch and the alignment angle of each optical component with periodic structure. To minimize the moiré effect, a moiré pattern pitch minimization method based on an analytic method in 3D display fields has been investigated [7–9]. This analytic method has shown good agreement if the transmitted intensity profiles through the optical components with periodic structure are similar. And the analytic minimization solution is provided between the equivalent grating of LCD and the special radial grating for pitch decision of parallax barrier in autosteroscopic display [10, 11]. However, if any of the transmitted intensity profiles through the optical components with periodic structure is distinctively different from others, the analytic method cannot provide an accurate estimation of the moiré effect. This is because the estimated maximum pitch length does not always have the highest visibility when the moiré patterns have various pitches, and the short periodic moiré patterns made a new moiré pattern with longer pitch.
To overcome the limitation of the analytic method in moiré pattern analysis, a bidirectional ray-tracing method has been adopted extensively because it is considered the most accurate method for prediction of the complicated moiré pattern . In spite of its accuracy, the bidirectional ray-tracing method has a disadvantage of long computation time due to a massive calculation involved in it.
Since the demand for high-resolution display devices, such as smart phones and notebooks is ever-increasing, design techniques to remove or reduce moiré images between the touch screen and panel are becoming more important [13, 14]. Such high-resolution display devices utilize very fine-pitched optical components. This inevitably imposes a massive load on optical scientists and engineers who try to optimize their display system within the limited time by using simulation code based on the ray-tracing algorithm [15–19]. Therefore, in this study, we propose a simulation algorithm for moiré patterns, which can greatly reduce the computation time while maintaining the accuracy. It will be shown that the new simulation tool can generate the moiré pattern using high resolution image process technique, and that the pitch and amplitude can be extracted quantitatively from the calculated moiré pattern.
2. Simulation algorithm and verification systems
The system is assumed to be made by the superposition of number optical layers which have periodic arrays with the same units as shown in Figs. 1(a) and 1(b). The intensity profile of the optical layer is the lateral intensity distribution of the unit area function which is an intensity profile defined by parameters along the local coordinate. This unit area function is defined by mathematical functions, or derived from the experimentally measured data and the simulation results using ray tracing. The intensity profile has accumulated intensity profiles from that of the light source layer to that of the optical layer. In other words, the intensity profile of optical layer represents the sum of the intensity through each optical layer from the emitting source layer.
The final image at the observation position is calculated from the convolution with the intensity profile at the optical layerand a point spread function (PSF) . Here, we define the PSF as an approximated two-dimensional Gaussian function of an Airy disc in consideration of the angular resolution of the human eye . The final spatial amplitude profile at the observation position is calculated as
After this procedure, it is possible to extract the moiré pitches and the amplitude per order from the frequency domain data, which is calculated by the Fourier transform from the sampling data of the wave vector direction of the moiré pattern. The moiré pitch per order is calculated asEqs. (1) and (2), we cannot only perform the simulation of the moiré pattern quickly but also analyze the precise pitch value of the main moiré pattern which has maximum visibility.
To verify the new algorithm, four cases of simulation were performed. In all simulations, an Intel Xeon computer with 2.93 GHz Quad core 2 CPU and 72GB RAM under MS Windows 7 64 bit OS was used. In the first case, a comparison between the new algorithm and the ray-tracing technique was carried out by using as optical system which had two optical layers with different periodic arrays. The computation time and quality of the image were compared. The second case was intended to make a direct comparison between the simulation and experimental measurement by using two brightness enhanced films (BEF), which were commercially available (3M Company) and used in LCD back-light systems. Thirdly, the usefulness of the new algorithm was tested by simulation of an optical system with high pixel resolution and a rather complicated pattern. For this simulation, a touch screen with a 5-inch diagonal panel with 800 x 1280 pixel resolution and a diamond-shaped periodic pattern was assumed. Finally, in order to verity the expandability of the proposed algorithm, a moiré simulation was carried out for a complex system in which two layers of lens-array patterns and a layer of black matrix panel are combined.
3. Results and discussion
Figure 2 shows the schematics of the evaluation system for the first case, which comprised a light source layer and two optical layers with different periodic arrays. The lower optical layer had bidirectional array pixels with 421.5 μm pitch, and the upper optical layer was a one-dimensional prism film which had apex angle of 90 degrees and 215 μm pitch. The refractive index of both layers was set to be 1.5 with dimensions of 50 x 50 mm2. The observation position was located at 50 mm from the upper optical layer. The light source, which was located beneath the lower optical layer, was defined as an emitting source with standard E spectral wavelength and Lambertian angular intensity profile . The output was assumed to be monitored by an image plane which had 5000 x 5000 pixels with resolution of 10 μm per pixel.
Figures 3(a)-3(c) compares the simulation results of the moiré patterns obtained from the bidirectional ray-tracing algorithm (Fig. 3(a)) and the new algorithm (Fig. 3(a)) by using the optical system shown in Fig. 2. Comparing the image qualities in Figs. 3(a) and 3(b), it is clear that the new algorithm can provide better image quality than the bidirectional ray-tracing algorithm. This is due to its ability to obtain the continuous intensity distribution of the image at the every point. In the case of bidirectional ray-tracing, if one wants to obtain image quality similar to that of the new algorithm, and a huge number of rays have to be used to fill every point. In addition to the moiré pattern image, it is possible to extract the main moiré pitch from sample data simultaneously. Figure 3(c) shows the extracted moiré pitches in the frequency domain, exhibiting the main pitch of 10 mm in this optical system.
Using the same computing hardware described in section 2, the calculation time needed for the ray-tracing method was 51.64 minutes, while that of the new algorithm was merely 0.08 minutes. This means that the computation time of the new algorithm is about 645 times faster than the ray-tracing method. This is because calculation with the new algorithm is based on the superposition of the intensity profile and the convolution with the spread function. On the other hand, the ray-tracing is based on calculation of the individual path of the wave through a system with regions of varying optical characteristics and reflecting surfaces . As a result, the computation time inevitably increases if the number of rays increases to obtain the accurate moiré pattern for a given system.
Moiré patterns were measured experimentally by using two commercially available brightness-enhanced films (BEF, 3M Company). Two sheets of BEF were stacked with precision angle alignment so that the angle between them might be 1° or 2°, and then pictures of the transmitted images were taken with a digital camera. Figure 4(a) shows an SEM image of the BEF which had an apex angle of 90° and pitch of 50 μm. Figures 4(b) and 4(c) show the moiré patterns obtained experimentally by using the sample size of 50 x 50 mm2 for the alignment angles of 1° and 2°, respectively. The measured moiré pitches were 2.86 and 1.43 mm for the alignment angles of 1° or 2°, respectively. When two gratings with the same pitch are aligned with angle 2α between them, the moiré pitch can be easily derived by the following Eq :
The exactness of the new algorithm was tested by using the BEF described above, and Fig. 4(d) shows the schematics of BEF used in the simulation. The dimension of the sample, apex angle, and the pitch were set to be the same as those of BEF used in the experiment. The simulated moiré patterns for the alignment angles of 1° and 2° are compared in Figs. 4(e) and 4(f), respectively. As can be seen from the measured and computed moiré patterns, the simulation results are in excellent agreement with those obtained experimentally. Also, the moiré pitches obtained from the simulation were 2.86 or 1.43 mm for the alignment angles of 1° or 2°, verifying the exactness of the new algorithm.
With recent developments in smart devices, the demand for touch screens with high precision is ever-increasing. In this study, moiré patterns, which can be generated from the combination of touch screen and panel with periodic structure, were simulated by using a touch screen with a metallic mesh of diamond-shape as depicted in Fig. 5(a). The width and thickness of the metal mesh were set to 10 and 1 μm, respectively. The line spacing between the mesh lines was 150 μm, and the acute angle between the crossed lines was 45°. This arrangement gives an effective horizontal (x-direction) pitch of 162.4 μm and an effective vertical (y-direction) pitch of 392 μm. An organic light-emitting diode (OLED) panel with periodic structure of 84.5 μm pitch, a black matrix of 5 μm width, and a Lambertian emitting angular profile were assumed to be positioned at 150 μm below the touch screen. The output images of the simulation had dimensions of 15 x 15 mm2 in which 7500 x 7500 pixels with 2 μm precision were arrayed. Figures 5(b) and 5(c) show the extracted moiré pitches in the frequency domain and the simulated moiré patterns obtained for the alignment angle between the touch screen and the panel of 0°. The moiré pitch with maximum intensity was 2.1 mm in this case. It is notable that this value of moiré pitch coincides exactly with that which would be obtained if the simulation were carried out by combining two linear grating patterns with 162.4 and 84.5 μm pitches. In the case of an alignment angle of 1°, the moiré pattern was tilted 26.7° with respect to the vertical axis. The moiré pitch was 1.9 mm. If one tries to simulate a moiré pattern for the same system using the bidirectional ray-tracing algorithm, the number of rays required to ensure accuracy would be very large. From a quantitative point of view, to obtain a moiré pattern with the same accuracy as that shown in Fig. 3(a), the number of rays would increase to the number of rays used in Fig. 3(a) multiplied by (10 μm / 2 μm)2. In other word, the number of rays should be 25 times larger than that used for the simulation shown in Fig. 3(a), which would certainly lead to tremendous computation time. This clearly indicates that the new algorithm will work as a useful tool for the simulation of moiré patterns for high-resolution optical systems with fine periodic pitch.
Figure 6(a) depicts the last simulation system with three periodic layers. The first layer was panel with BM (black matrix) whose pitch and width is 295 and 50 μm, respectively. The second and the third optical layers are composed of micro convex lens array on absorptive base film with slightly different dimensions. The width and the radius of curvature of the second layer were set to be 225 and 315 μm, respectively. Due to the hexagonal arrangement of the lens array, the x- and y-directional (horizontal and vertical direction in Fig. 6(a)) periods were different. The x-directional pitch was 315 μm and y-directional pitch 273 μm. The third optical layer had lens array on absorptive base film with 245μm width and 335 μm radius of curvature, and the periods of x- and y-direction were 335 and 290 μm, respectively. For each layer, the refractive index was set to be 1.5 with dimensions of 15 x 15 mm2. A light source layer was located beneath the panel layer with the same properties as described in the first case shown in the Fig. 2(a). The output images of the simulation had 7500 x 7500 pixels with 2 μm precision. The simulation results of the moiré patterns obtained from the bidirectional ray-tracing algorithm and the new algorithm are compared in Figs. 6(b) and 6(c), respectively. As was the case of the calculation with two periodic optical layers, the new algorithm could provide similar or better image quality than the bidirectional ray-tracing algorithm. Figures 6(d) and 6(e) show the extracted moiré pitches in the frequency domain for the x- and the y-direction, respectively. The computation time needed for the ray-tracing method and the new algorithm method were about 148 minutes and 11 seconds, respectively. The calculation speed of the new method was about 807 times faster than that of the ray-tracing method. The more pronounced difference in simulation speed than the first example with two periodic optical layers is due to the difference in complexity of the system. The above results clearly tell that the new algorithm can afford to deal with complex system efficiently, and that great reduction of the simulation time can be attained.
Lastly, it has to be mentioned that, although our new algorithm can provide much faster computation time and better image quality than the bidirectional ray-tracing method, it has its own limitation. Unlike the bidirectional ray-tracing method, the new algorithm cannot deal with the situation in which any complex optical property or complicated surface morphology is engaged in the system. However, since the computation speed of the new algorithm is unparalleled, the time-consuming procedure of moiré period minimization can be achieved by a combination of the new algorithm with the ray-tracing method. For instance, if an optical component with periodic structure is comprised of scattering material, one can determine the moiré pitch and angle instantly by carrying out the simulation while assuming that the material is free from scattering. Since the scattering effect does not affect high frequency moiré intensity, the new algorithm will provide a rough insight into the moiré pitch, angle, and intensity. After this, one can obtain the precise moiré pattern with the aid of the bidirectional ray-tracing technique.
This paper proposed a new computational algorithm for simulation of the moiré patterns based on convolution with the superposition of the intensity profile transmitted from the optical layers and the point spread function. Comparison with the bidirectional ray-tracing method revealed that the new algorithm could provide much faster computation time and clearer image of the moiré pattern. The validity of the new algorithm was double-checked by comparing the simulated results with experimental results. It was also shown that the moiré pitch could be extracted from the sampling data simultaneously, and the exactness of the derived moiré pitch was verified by comparison with that obtained by analytical solution. The usefulness of the new algorithm was also tested by using a complicated optical system comprised of a touch screen with a diamond-shaped metal mesh and a panel with fine periodic structure. Finally, the efficiency and the expandability of the new algorithm was verified by simulating a three-layered system with two lens array layers and one BM panel, and by making comparison the results with those obtained by the ray-tracing technique.
This work was supported by the Korea Institute of Science and Technology (KIST) Institutional Program (Project No. 2E24012).
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