1. Introduction

A stereoscopic display can give people a feeling that cannot be obtained from a 2D display. To realize it, two images need to be shown to two eyes respectively. Auto-stereoscopic televisions and mobile displays based on spatial multiplexing have been realized with fixed lenticular lens arrays or parallax barriers [1,2]. However, there is a serious restriction of such an approach, as to put two images on the display panel at the same time, reducing image resolution by half [3,4], which makes it difficult to be accepted by general public. We propose to use time multiplexing of full display resolution images to different directions to achieve the same stereo effects with the full resolution. This can be achieved by using high quality, pixel-level, switchable liquid crystal (LC) micro-lenses [5].

The development of LC micro-lens technology has enabled switchable auto-stereoscopic display to be realized in recent years [6–8]. The electro-optical nature of LC lens allows the viewing mode to be switched between 2D and 3D if needed such as when viewing text contents [9,10]. The reported size of the demonstrated LC lenses, defined here as the short length of the circular/lenticular configuration, ranges mostly between a few hundred of micrometers and a few tens of millimeters [11–17]. Such sizes are suitable for modern desktop displays but not for mobile display pixel sizes. Because of the large lens size, the LC layer is also very thick, a few tens of micrometers, even for a high birefringence LC. The resultant high response time of LC, in the scale of seconds, will therefore prohibit the development of the time-multiplexing scheme in auto-stereoscopic displays.

The LC lens size of a few tens of micrometers was designed, fabricated and demonstrated to show image steering ability with high quality phase profiles [5]. However, there are a number of questions regarding the performance and effect of these LC lenses which remain to be answered, such as steering angle for comfortable viewing distance, crosstalk and scalability [18]. Simulation results of less than 10% crosstalk have been reported, but the light source (pixel or sub-pixel intensity distribution) is often un-documented [6] and a perfect LC lens phase profile is assumed [19,20]. For practical purpose, the simulation and measurements need to be done with both the measured input light source and the measured LC lens phase profile. Without these work done, it is very difficult to apply and develop LC lens technology on modern mobile phones for the high quality auto-stereoscopic viewing experience.

There have been various attempts in increasing the resolution of auto-stereoscopic displays including “movable or scanning” LC lens with multi-electrode driving configurations, but they were all at concept level and could not manage to have the same resolution as the display panel. In this study, we demonstrate an auto-stereoscopic display based on the switchable LC micro-lens array on a full size mobile display. Micro-lens array over the full size of a mobile display is designed, fabricated and assembled onto a high resolution smart phone display panel. Characterizations are carried out to evaluate the optical performance and the factors for improvements. Simulation models are set up to further optimize the performance, including the steering angle and crosstalk using the measured sub-pixel spatial light intensity distribution of the mobile display and identify the conditions for achieving satisfactory performance.

2. Full resolution auto-stereoscopic display and device design and fabrication

A multiplexing scheme using interlaced spatial and temporal beam steering for auto-stereoscopic displays was proposed [21]. The detailed characterisation of the LC lenticular phase lens structure and its phase profile were reported previously [5]. In short, every such a lenticular lens covers two neighbouring columns of sub-pixels on the display panel’s colour filter (CF) layer, and each lens will be able to steer the light from the two columns of sub-pixels to different eyes, respectively, as shown in Fig. 1(a). For the subsequent image frame, the whole phase profile for the lens array is shifted by one sub-pixel, as shown Fig. 1(b), in order to steer the light from the same sub-pixels to the direction opposite to that in the previous frame. Due to the residual viewing effect of human eyes one will be able to “see” the two time sequential images as a single image with the same image resolution as that of the display panel itself.

 

Fig. 1 The proposed multiplexing scheme using interlaced spatial and temporal beam steering to obtain the maximum resolution for an auto-stereoscopic display. The multiplexing operation can be described in two steps: (a) Frame 1 – odd sub-pixels deflected to the right eye and even sub-pixels deflected to the left eye. (b) Frame 2 – odd sub-pixels deflected to the left eye and even sub-pixels deflected to the right eye. The phase pattern is shifted by one sub-pixel resulting in the swap of steering directions.

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The large scale LC lenticular micro-lens devices with glass substrate dimension of 60×80 mm² (4 inch diagonal) are made in-house in a cleanroom environment. They consist of an 8 µm thick nematic LC layer sandwiched between two glass substrates. The thicknesses of substrates are 0.15 mm (Schott AF32 eco Thin Glass, Schott, Mainz, Germany) and 0.55 mm (CEC100S, Präzisions Glas & Optik GmbH, Iserlohn, Germany), respectively. The glass surface that is in contact with the LC layer of both substrates has an ITO layer with a sheet resistance of 100 Ω/□. The ITO layer on the 0.55 mm thick glass substrate was patterned with photolithography and chemical etching to form line electrodes which are between columns of the sub-pixels of the mobile display as shown in Fig. 2. The electrode tracks controlled by V₁ and V₂ are interlaced, they are switched on in sequence to shift the phase pattern by one sub-pixel to result in the swap of steering directions.

 

Fig. 2 The LC micro-lens array design (top-down view) intended to steer images at sub-pixel level on a commercial mobile display for (a) vertically handheld orientation and (b) horizontally handheld orientation. The size of electrode tracks and spacing are not in scale.

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The commercial mobile display used for demonstration is supplied by BOE (BTL454885-W578L R0.2, BOE Technology Group Co., Ltd, Beijing, China), it is a LC flat panel display (FPD) with a resolution of 854 × 480. The sub-pixel of the mobile display is 30.5 µm wide and 100 µm long. Each pixel, consisting of red, green and blue sub-pixels, has a dimension of 115.5 × 115.5 µm². The sub-pixels have a slanted configuration with about 8° angle from the vertical as shown in Fig. 2(a) inset, and they overlap with the patterned electrodes. The polarisation direction of the mobile display is along its longer side.

The line electrodes are parallel to the polarisation direction of the mobile display for the vertically handheld orientation and perpendicular to it for the horizontally handheld orientation. The ITO layer on the 0.15 mm thick glass substrate is used as a common electrode. A homogeneous polyimide alignment layer (AL-3046, JSR Co., Tokyo, Japan) was spin coated and rubbed in an anti-parallel configuration along the longer side of glass substrates.

The assembled LC devices were placed directly over the entire mobile display with the device rubbing direction matching the display polarisation direction. An index matching liquid (IML 150, Norland Products Inc., Cranbury, NJ, USA) was applied in-between the LC device and the display surface to eliminate the refractive index difference caused by an air gap. The index matching liquid has a similar refractive index (1.52 @589nm) to typical glass substrates. When a voltage of 1 kHz square wave and 5 V rms is applied to electrode V₂, LC lenticular lens array is formed as shown in Fig. 3(a). It is indicated by the bright and dark fringes close to the electrode tracks when the device is observed under microscope at a 45° angle between cross polarisers. The measured light intensity data is converted to phase values and the results are plotted for two wavelengths at 510 nm and 635 nm as shown in Fig. 3(b). The LC phase profiles have abrupt jumps because of the measurement setup and calculation methods used, which were discussed in details in the previous study [5]. The phase profiles are used to simulate the LC lens function in section 4 and the shape of measured profiles are discussed in section 4 and 5.

 

Fig. 3 (a) Auto-stereoscopic display configuration with the LC lens layer, indicated as an intensity image of phase lenticular structure driven by electrode V₂, over the mobile display, indicated as pixel CF layer. (b) Calculated LC lens phase profiles at 510 nm and 635 nm from the measured intensity data.

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3. Demonstrators and performance characterisation

3.1 Full size demonstrators

The LC lens device fabricated according to Fig. 2(a) is demonstrated with the display held vertically as shown in Fig. 4(a), the ultra-thin glass (0.15 mm) is in contact with the mobile display. Another LC lens device fabricated according to Fig. 2(b) is also demonstrated with the display held horizontally as shown in Fig. 4(d), the thicker glass (0.55 mm) in contact with the display. All the images are captured with a SLR camera (Nikon D7000).

 

Fig. 4 (a) Rendered auto-stereoscopic image when the mobile display is held vertically and the LC lens is off. Steered images (cropped) to the left eye (b) and right eye (c) from the rendered image in (a) when the LC lens is switched on. (d) Rendered auto-stereoscopic image when the mobile display is held horizontally and the LC lens is off. Steered images (cropped) to the left eye (e) and right eye (f) from the rendered image in (d) when the LC lens is switched on.

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Auto-stereoscopic images were rendered with MATLAB and displayed for both display orientations. The rendered image with the LC lens off and steered images with the LC lens on are captured and shown in Fig. 4. When the LC lens is off, the rendered auto-stereoscopic image in Fig. 4(a) consists of two stereo images, one for each eye. The left eye image is rendered with odd columns of sub-pixels and the right eye image is rendered with even columns of sub-pixels. When the LC lens is switched on, each stereo image is steered to the corresponding eye of the viewer, as shown in Fig. 4(b) and (c) respectively. The image steering is obvious and only contains the information intended for the target eye. For example, the bird’s eye in Fig. 4(a) and (d) shows a double image because it contains information for both eyes before the LC lens is switched on. In contrast, the bird’s eye in the steered images shown in Fig. 4(b), 4(c), 4(e) and 4(f) only show a single crisp black dot because the LC lens steered the information for one eye from the other.

In both orientations, the LC lenticular lens array was able to separate the image for one eye uniformly from the other and the viewer is able to enjoy the auto-stereoscopic experience. The steered images show no distortions from colour separations and the LC lens mis-alignment. The thin coloured vertical strips in Fig. 4(b) and 4(c) are caused by the missing ITO electrodes which have been over etched.

The vertically handheld demonstrator delivers a 2.8° steering angle and 600 mm optimum viewing distance. In comparison, the horizontally handheld demonstrator delivers a 5.5° steering angle and 350 mm optimum viewing distance. The difference in the steering angle is a result of sub-pixels having a rectangular shape, with its length three times larger than its width.

3.2 Full resolution demonstration

The test image is rendered with every other red and green sub-pixels illuminating light beams as shown in Fig. 5(a.i) and the captured image is shown in Fig. 5(a.ii) when the LC lens array is off. When the LC lens is switched on with voltages applied to V₂ not to V₁, as shown in Fig. 3(a), the LC lenticular lens array forms and the green and red light beams are steered towards the viewer’s left eye and right eye separately as shown in Fig. 5(b.i). Each of the viewer’s eye sees half of the display resolution. The steered green image is captured and shown in Fig. 5(b.ii) with the camera at the left-eye-location.

 

Fig. 5 (a) Demo configuration and a test image when the LC lens is off. (b) Demo configuration and a steered image to the left eye when the LC lens is on with voltages applied to V₂. (c) Demo configuration and a steered image to the same eye when the LC lens is on with voltages applied to V₁.

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When the voltages to V₂ is switched off and V₁ switched on, the LC lens array is shifted by one sub-pixel. The green light beam in this case is steered towards the viewer’s right eye and the red light beam towards the left eye respectively as shown in Fig. 5(c.i), the steered directions are different from the previous case. The steered red image is captured and shown in Fig. 5(c.ii) with the camera at the left-eye-location. Each of the viewer’s eye sees the other half of the display resolution. With the voltages switched between V₁ and V₂ frequently, the steering angle swaps at the same rate. As a result, all of display pixels are steered to both of viewer’s eyes and a full resolution auto-stereoscopic mobile display is demonstrated.

3.3 Characterisation setup

The demonstrated auto-stereoscopic display is characterised in terms of the steering angle and crosstalk. The characterisation setup was that the mobile display with the attached LC device was held vertically, the display surface was perpendicular to the optical table and it was 600 mm away from an optical rail as shown in Fig. 6. The optical rail was parallel to the display surface. The camera was mounted onto the optical rail so the steered light intensity image could be captured at different locations along the rail in the viewing plane. The movement range for the camera was 200 mm with an interval of 10 mm between two neighbouring locations. When camera moved to a new location, it was rotated slightly so that the testing image always appeared in the centre of the captured image.

 

Fig. 6 Schematics of the steering angle and crosstalk measurement setup.

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The measurements were carried out in a dark room condition. The displayed black image was captured and camera settings were calibrated so the black image intensity was zero. A black masking paper was used to cover the display apart from the testing area to avoid light leakage from other part of the display. Another black masking paper was fixed in front of the camera lens with a circular aperture cut in the middle to mimic the human pupil, the aperture size was ~6 mm. The testing area on the display and the aperture of the camera were of the same height from the optical breadboard.

Test images consisting of single colour and black line pairs are shown as insets in Fig. 7. Every other columns of red or green sub-pixels illuminate light so the single colour is always steered to one direction for the measurement, red image to the right eye and green image to the left eye as shown in Fig. 7(a) and 7(b), respectively. In the measurement, only one colour from the display is switched on for each experiment, this is to ensure that the non-linear sensitivity of the camera sensor at different colour bands does not affect the results.

 

Fig. 7 Crosstalk measurement configurations (a) when every other columns of red sub-pixels are illuminating light and (b) when every other columns of green sub-pixels are illuminating light.

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The steered colour intensity distribution image at different locations along the optical rail were captured when the LC lens array was switched on and off for comparisons. The same camera settings were used for all the captured images, which were then analysed with MATLAB. The intensities were averaged over an area of 5×5 mm² on the display.

An ideal auto-stereoscopic mobile display possesses two essential features; a comfortable viewing distance and a minimum crosstalk. The comfortable viewing is achieved through a sufficiently large steering angle ($\alpha$) for each eye to see the highest intensity steered beam at an average handheld distance, 300-350 mm. The steering angle ($\alpha$) needs to be 5-6° based on an average inter-pupil distance of 65 mm. It is determined by distances ($d_x$) and ($d$) with a relationship of $\tan (\alpha )=d_x/d$ as shown in Fig. 7(a), ($d_x$) is the horizontal distance between the centre of the sub-pixel and the centre of the corresponding LC lens and ($d$) is the separation distance between the LC lens layer and the pixel CF layer.

The definition of the second feature, crosstalk, is the percentage intensity ratio of ghost image observed at one eye to the correct image observed at the other eye and the equation is written as

3.4 Steering angle

The steering angle ($\alpha$) was measured based on the location of the highest intensity image at the viewing plane for two distances ($d$ = 0.4 mm and $d$ = 0.8 mm), consisted of the mobile display cover glass (~0.25 mm thick) and LC lens glass substrates (0.15 mm and 0.55 mm) as shown in Fig. 7(a), and the results are 2.8° and 1.2° respectively. By having a shorter separation distance ($d$ = 0.4 mm), the viewing distance of the auto-stereoscopic display has been reduced to 600 mm from 1,400 mm at $d$ = 0.8 mm. The results satisfy the trigonometrical relationship between ($\alpha$), ($d_x$) and ($d$). To further reduce the viewing distance, we need a larger ($d_x$) value or a smaller ($d$ value. The distance ($d_x$) is effectively half of a sub-pixel pitch, in the range of a few tens of micrometres, and it is getting smaller in modern mobile displays. So the separation distance ($d$) needs to be reduced, to a value of around 0.2 mm for a comfortable viewing experience, this is discussed in section 5.

3.5 Crosstalk analysis

To analyse the crosstalk, intensity values at a fixed plane parallel to the display surface are measured at each location and normalised and plotted for red and green separately in Fig. 8(a) and 8(b). The $x$ = 0 mm in both figures is the centre line between the viewer’s eyes. Note that the points at this fixed plane do not have the same viewing distance to the centre of the mobile display. When the LC lens was off, the normalised intensity shows a peak value in the centre, indicated by the fitted curve in Fig. 8(a), it then decreases as measured points are further away from the centre. The intensity variation is within 20% across a 150 mm range, showing a large viewing/diverging angle of a typical FPD. The intensity distribution for green test images is very similar to red ones with the LC lens off, but the variation is less, within 10% across the same distance range.

 

Fig. 8 Intensities of the (a) red and (b) green colour images when camera was moving along the optical rail in the viewing plane. Inset images indicate the calculated area of the highest and lowest intensities for both red and green when the LC lens is on.

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When the LC lens was switched on, the tested colour image was clearly steered to one direction, resulting in an increased intensity at one side of the centre location and a decrease at the other, shown as inset images in Fig. 8(a) and 8(b) respectively. Steered images with the highest intensity, are about 30 mm from the centre, similar to half of the average inter-pupil distance of human eyes. The observing distance of 600 mm makes it possible to view an auto-stereoscopic display on a mobile device.

The crosstalk for the red image is 26%, calculated according to the Eq. (1). The result is not ideal and the main reason is thought to be a large viewing/diverging angle of light beams from display pixels, shown in Fig. 8 as the measured intensity values when the LC lens if off. As most display manufacturers specify, the viewing angle for modern FPDs are approaching 85-90° half angle with an aim for multiple viewers to see it from different angles without much brightness reduction. This is done by integrating a scattering layer to FPDs at the light exit. This wide viewing angle however, makes it very challenging for auto-stereoscopic applications. A large portion of sub-pixel light beam goes through neighbouring LC micro-lenses as a result, which causes large crosstalk and diminish auto-stereo effects.

The rubbing quality of the LC device alignment and the extinction ratio of the display polarisation layer are also thought to contribute to high crosstalk, causing some of the light beam from the display not being steered. In addition, the mobile displayed used has the tilted pixel orientation, which means the illuminated light beam go through more LC lenses than vertical orientated pixels and are not steered to the intended direction. Furthermore, the LC lens phase profile as shown in Fig. 3(b) is not a perfect parabolic shape, the crosstalk with the LC lens is compared to an ideal lens in section 4.4

The crosstalk for the green image is 41%, larger than the red image value. Two main factors are thought to contribute to the difference. The first one is that the green image intensity distribution has a larger viewing/diverging angle when the LC lens is off, as shown in Fig. 8(b) compared with Fig. 8(a). Over the same distance range, the green intensity only drops ~10% from its peak in the centre while the red intensity drops twice as much. As a result, more green light goes to the neighbouring LC lenses comparing with red and produces more crosstalk. This is to do with the display CF scattering layer having wavelength dependent properties. The second one is that the effective focal length (EFL) of LC lens at red wavelength (0.36 mm) matches the separation distance $d=$ 0.4 mm) better than at green wavelength (0.34 mm), so the steered green light is more spread than the steered red light at the viewing plane, resulting in a higher crosstalk. The crosstalk at the separation distance $(d=$ 0.8 mm) was also measured and it is about 70%, higher than the case of a shorter separation distance.

3.6 Display pixel intensity distribution

In our study, the half viewing angle of the mobile display was measured using a microscope (Olympus BX51, Olympus, Tokyo City, Tokyo, Japan) that focuses at different layers including and away from the pixel CF layer. The captured images by the colour camera (Retiga 4000R CCD Camera, QImaging, Surrey, BC, Canada) show the spatial light intensity distributions of the red sub-pixel at different layers that are parallel to the pixel CF layer, shown in Fig. 9(a). Every other columns of red sub-pixels were switched on for the measurement.

 

Fig. 9 (a) Microscope images of red sub-pixels at and away from the pixel CF layer. (b) The spatial light intensity distribution of the red sub-pixel of the corresponding images in (a). (c) The peak intensities and the profile width (FWHM) as a function of the distance away from the pixel CF layer.

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Horizontal line profiles, each averaged over several lines of data across the measured light intensity image and plotted for each corresponding layer respectively, are shown in Fig. 9(b). The relationship of the peak intensities of the red sub-pixel and the full-width-half-maximum (FWHM) of intensity profiles with respect to the variable separation distance away from the CF layer is extracted from Fig. 9(b) and plotted with curve fittings in Fig. 9(c), respectively. The peak intensity of the red sub-pixel, indicated as red triangles, decreases exponentially with the separation distance. The FWHM of the intensity profile, indicated as dark blue solid circles, increases linearly with the separation distance. The half diverging angle of the mobile display is about 15°. The pixel light intensity distribution of a different commercial mobile display (Huawei Mate I, Huawei Consumer Business Group, Shenzhen, Guangdong, China) was also measured and the result is similar to the BOE mobile display.

With a short separation distance, the sub-pixel light intensity available for the LC lens steering will be higher and the FWHM of intensity profile will be narrower at the LC lens layer. It results in a lower crosstalk as measured values indicate in section 3.5 and brighter auto-stereo images. The section below further investigates and optimises the LC lens array performance based on the measured display pixel intensity distribution.

4. Simulation analysis

This section evaluates performance of the auto-stereoscopic display configuration by modelling the experimentally measured light source and using measured LC lens profile as shown in Fig. 3(b) in the simulations. It includes details of how the physical setup is modelled and wave propagation based simulations are run. One of the key challenges was to model the source radiation intensity pattern. It significantly affects the intensity profile of light at the LC lens layer, as well as crosstalk, steering angle and width of the propagated light at the viewing plane. The input source consists of LED backlight and pixel CF layer while it is still possible to model the light propagation as a monochromatic light source.

4.1 Model configuration

The simulation setup is modelled as shown in Fig. 10. The width of the sub-pixel is px = 30.5 µm, and the half width of LC lens is D/2 = 38.5 µm. The 4 µm half width ITO electrodes reduce the half width of LC lens to L/2 = 34.5 µm. The separation distance, d, was simulated for the values 0.4/0.8/1.4 mm whereas the viewing distance, z, from the LC lens layer to the viewing plane, was chosen as 300/600 mm. The focal length, f, of LC lenses was set to the values, which are calculated from the experimental data, 0.36, 0.34, 0.29 mm for red (650 nm), green (550 nm) and blue (450 nm) colour channels, respectively. Green and red wavelengths were simulated to compare with the experimental results.

 

Fig. 10 Simulation setup and the system model for wave propagation.

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The definitions of s(x), $h_{\alpha }(x)$, $d(x)$, $LA(x)$, $h_{\beta }(x)$, ${|.|}^2$ and I$(x)$ and their relationships are described in the subsequence text. It is assumed that all the light from the pixel CF layer is modulated by LC lenses, which means the display light source is perfectly polarised and the LC lens device has a perfect alignment layer with its rubbing orientation along the polarisation direction. The simulation does not take into account the tilted pixel configuration.

4.2 The light source in use

The source function, s(x), is modelled such that a similar intensity distribution having the same FWHM as the measured values shown in Fig. 9(b), is obtained when the source function propagates to the LC lens layer. For this purpose, a Gaussian basis function, $g(x)=exp(-a{x}^2)$, was chosen. However, the wave propagation of a single Gaussian function at a given separation distance does not reach the same FWHM and give a similar intensity profile as the experimental data. Therefore, the source function is modulated by multiplying a finite number of weighted superposition of complex sinusoids, $w\left(x\right)={\displaystyle \sum }_{k=1}^T{c}_k{\text{φ}}_{\text{k}}(x)$, where ${\text{φ}}_{\text{k}}(x)=exp(j\frac{2\pi }{\lambda }\sin ({\text{θ}}_k)x)$ and the constants $c_k$ are chosen from another Gaussian distribution and the divergence angle parameter ${\text{θ}}_k$ is ranged between {−25,25} degrees with 1° angular spacing. This way, it is aimed to control the spread of the source function as it propagates the glass medium. The function, $w(x)$, is actually a summation of plane waves directing at discrete angles with varying amplitudes. The centre coefficient is set to a value of 1 and decreases towards 0 at higher angles. The resulting source function is given as $s\left(x\right)=w\left(x\right)\times g(x)$. The source function is then propagated to lens array plane as $d\left(x\right)=s\left(x\right)*h_{\alpha }(x)$, Fig. 10, where $h_{\alpha }(x)=\text{exp}(j\text{α}{x}^2)$, with $\text{α}=-\frac{\pi }{\lambda d}$, is the kernel function of the wave propagation defined between the pixel CF layer and the LC lens layer. The asterisk, $*$, is the convolution operation. The wave propagation can be rewritten as $d(x)=h_{\alpha }(x)\times ℱ\{s(x)\times h_{\alpha }(x)\}$ where $ℱ\{·\}$ is the Fourier transform operator. The Fourier transform part shows how the propagated field is spread at the lens array plane as follows

 

Fig. 11 Shifted Gaussian basis functions with different amplitudes at the lens array plane represents the spread of light intensity over a certain width.

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The intensity at the LC lens layer is given as ${\left|d(x)\right|}^2$. The obtained intensity distribution of the simulated results are plotted in Fig. 12 at several separation distances. Intensity profiles and calculated FWHMs and peak intensity ratios of the simulated light source at LC lens layer match that of experimental values in Fig. 9(b) very well. After the light reaches the LC layer, the complex amplitude of the propagated field, $d(x)$, is multiplied by the lens array phase profile, $LA\left(x\right)=L\left(x\right)*{\displaystyle \sum }_{r=1}^R\delta (x-rD)$. The resultant field is further propagated to the viewing plane by convolving with $h_{\beta }\left(x\right)$ where $\beta = -\frac{\pi }{\lambda z}$. The magnitude square of the complex field will give the intensity distribution, $I(x)$, at the viewing plane [22].

 

Fig. 12 Simulated light intensity distribution of display sub-pixel at the LC lens layer with different separation distances. The intensity curves for each colour are normalised to the maximum value at which the simulation was done when $d=$ 100 µm. The simulated sub-pixel light intensity distribution according to the red colour channel is used for simulation at all colour channels.

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In the physical setup, LC lenses are made between glass substrates. So, in the simulations, the input is first propagated by setting the refractive index n = 1.5, which modifies wavelength parameter $\to \lambda /n$, then it is multiplied by the lens phase. Afterwards, it is propagated in the free space, n = 1, to obtain the output intensity distribution.

The simulations are performed for each “on” sub-pixel separately, and then, each resultant intensity distribution at the viewing plane are superposed as

4.3 Micro-lens phase profiles

The ideal lens profiles are generated for the given parameters as described in Section 1.1 for both red and green wavelengths. In general, an ideal thin lens profile, $L\left(x\right)$, can be regarded as the conjugate of kernel function, $h_{\gamma }\left(x\right)=exp(j\gamma x^2)$ with $\gamma =- \frac{\pi }{\lambda f}$ . The free space propagation $L\left(x\right)*h_{\gamma }(x)$ should give an impulse function, $\delta \left(x\right)$, at a distance f. Therefore, $L\left(x\right)$ is defined as $h_{\gamma }^{*}\left(x\right)$ with $\gamma =\frac{\pi }{\lambda f}$. The continuous function that represents the LC lens can be regarded as a truncated lens function, i.e. $rect\left(x\right)=1, x\in [-1/2,1/2]$ where $L\left[n\right]=L\left(nX\right)=exp(j\frac{\pi }{\lambda f}{n}^2{X}^2)$ and 0 elsewhere. The sampled discrete lens function [22,23], used in the simulations is given as $L\left[n\right]=L\left(nX\right)=exp(j\frac{\pi }{\lambda f}{n}^2{X}^2)$, for $n\in \{-L/2X, L/2X-1\}$ where the $[\cdot ]$ operator rounds to the nearest lower integer value. In our case, the discrete lens function was modelled such that width of ideal lens and peak phase level values of the unwrapped profiles matches the measured LC phase profile which is shown in Fig. 13. For our simulations, in order to have same amount of samples for measured data as the numerical data, we had to down-sample measured data by decimating it by a factor of, $3\downarrow$, and passing through a simple low pass filter, $lpf\left[n\right]=\{0.25, 0.5, 0.25\}$. The down-sampling operation smoothens the data while retaining the shape of profiles and phase depth. The LC phase profiles were measured at 635 nm and 510 nm, respectively, slightly different from the simulated wavelengths of 650 nm and 550 nm of the display. The measured phase values are scaled to the simulated values according the ratio of the maximum phase depth in these two cases. As a result, the maximum phase depths become 6.3 π and 8 π at 650 nm and 550 nm respectively. There are discontinuities and non-steering region as shown in Fig. 13(b). Some of them are measurement artefacts and some can be further improved (see section 5.2).

 

Fig. 13 Unwrapped phase levels for (a) generated ideal lens profiles and (b) measured LC lens profiles for red and green colours.

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4.4 Analysis of the crosstalk generated by multiple source pixels

The simulations were designed to be able to evaluate the performance of the physical LC lenses in terms of the ability of light steering and amount of crosstalk introduced. The simulation results of the ideal and LC lens are compared.

The number of LC lenses are set to 61 whereas the number of sub-pixels to 19. The “on” sub-pixels, which are shown as solid rectangles in Fig. 10, illuminate light beams through the LC lenses. Every other red or green sub-pixels are switched on, respectively. The ideal lens profiles were simulated to set the reference. Figure 14 shows cross-sections of simulated intensity distributions of steered red and green sub-pixels with ideal lenses at the viewing plane, with different separation distances $d=$ 0.4, 0.8 and 1.4 mm and $z=$300 and 600 mm. In order to calculate the steering angle, ${\vartheta }_s$, the distance, $x_d$, from the peak intensity location to the centre point between two eyes is calculated. Then this is simply converted to angle values by calculating ${\vartheta }_s=a\tan (x_d/z)$. For the crosstalk, we calculated the ratio of average intensities of light over an average pupil size (6mm) received by each eye for the same colour channel. For example, to calculate the red channel crosstalk, the average intensity value of the light received by one eye $A={\displaystyle \sum }_{pupi{l}_{left}}{I}_R(n)/pupi{l}_{left}$is divided by the average intensity value of the light received by the other eye $B={\displaystyle \sum }_{pupi{l}_{right}}{I}_R(n)/pupi{l}_{right}$. Then, the crosstalk value is given by $100\%\times A/B$. In this example, we assume that the value of A is corresponding to the ghost image appearing at the left eye location and the value of B is corresponding to the correct image appearing at the right eye location.

 

Fig. 14 Cross-sections of simulated intensity distributions of steered red and green sub-pixels with ideal lenses at the viewing plane, at various distances (d) and (z).

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The calculated results are summarized in Table 1, a shorter separation distance ($d$) value results in an increase in the steering angle and a decrease in the crosstalk at the viewing plane, when measured at $z=$600 mm. When the viewing plane is at $z=$300 mm, the crosstalk for longer separation distance ($d=$0.8 or 1.4mm) is more than 100%, resulting in a pseudo-stereoscopic image. The largest steering angle and lowest crosstalk are obtained for $d=$ 0.4 mm. In addition, the EFL of the lens is another important contributing factor for the crosstalk reduction. When the EFL approaches the separation distance ($d=$0.4 mm), the lens will generate a most collimated light with minimum spread. Hence in principle when the EFL reaches the separation distance the crosstalk becomes the lowest.

Table 1. Simulated steering angle and crosstalk values at d = 0.4, 0.8 and 1.4 mm and z= 300 and 600 mm with ideal lenses.

View Table

The same set of parameters were used for simulations with the LC lens array, which the intensity distributions of steered red and green sub-pixels at the viewing plane are plotted in Fig. 15. The results at separation distances $d=$ 0.8/1.4 mm are not presented due to the low steering angle and high crosstalk. The steering angle at $d=$ 0.4 mm matches the ideal lens case, but the crosstalk is higher, 34-35% at $z=$300 mm and 19-22% at $z=$600 mm, about three times higher than the ideal lens case. The imperfect LC phase profile is thought to be thecause of the higher crosstalk. Comparing with the ideal lens profile, the LC phase profiles have discontinuities on the phase slopes as shown in Fig. 13(b), and there is a large portion (14-20 µm wide) at the top of profile which is flat and does not steer beam. The LC lens phase profile can be optimised and this is discussed in Section 5.

 

Fig. 15 Cross-section of simulation results at the viewing plane of red and green sub-pixels and measured LC phase lenses at the separation distance $d=$ 0.4 mm with different viewing distances (a) $z=$ 300 mm and (b) $z=$ 600 mm.

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The simulated crosstalk based on the LC lens is in a good agreement with the measured one (26%) at red wavelength since the pixel intensity distribution is simulated according to the red colour channel. The simulated steering angle is higher than the measured one (2.8°) at $d=$ = 0.4 mm. It is thought that the index matching liquid layer has a finite thickness and the separation distance is slightly larger than 0.4 mm in the experiments.

After obtaining the intensity distributions at fixed separation distances, simulation is also done at varying viewing distances ($z=$ 0-600 mm) in order to picture the cross-section of the light field along the propagation axis from the CF layer to the viewing plane. Figure 16 shows the spatial light intensity distribution without lens and with the ideal and LC lenses respectively. The LC lenses are not steering the light as perfectly as the ideal lenses, the image in Fig. 16(c) shows a more smeared light distribution with less pure colour at each eye location than the image in Fig. 16(b). However, the amount of light reaching to each eye is still sufficient to separate the red and green colour channels. In addition, there is a large range of depths where viewers can see auto-stereo images, from 300 to 600 mm, although the ‘sweet spot’ is at around 450 mm.

 

Fig. 16 Spatial light intensity distribution of the simulated light source with (a) no lens, (b) ideal lenses steering and (c) LC lenses steering in front of the pixel CF layer at 0-600 mm propagation distance. Magenta circles indicate viewers’ eye locations.

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5. Further optimisation and improvement

5.1 Optimised configurations

Both the measured and simulated results show a better LC lens performance, a larger steering angle and a lower crosstalk, with a reduction of the separation distance. The geometrical relationship is plotted in Fig. 17, the steered images at the measured steering angle are shown as insets in Fig. 17 at different separation distances ($d=$ = 0.8 and 0.4 mm). The red and green letters (R & G) image was used for the comparison, one letter was steered to one eye.

 

Fig. 17 The steering angle and the viewing distance variation with respect to the separation distance ($d$) with inset steered images at the distances $d$ = 0.4 and $d$ = 0.8 mm, respetively.

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The decrease in the separation distance clearly has a positive impact not only on the steering angle, but also on the steered image, resulting in better perceived images with low crosstalk for the auto-stereoscopic effect. For example, the steered letter ‘R’ at d = 0.4 mm shows a much more dominant red colour than the one at $d=$ = 0.8 mm.

The viewing distance of the auto-stereoscopic display has been reduced from 1,400 mm to 600 mm as a result of the shorter separation distance ($d=$ 0.4 mm). It is still a few hundred millimetres away from the optimum range of 300-350 mm for mobile display applications as shown in Fig. 17, this cannot be achieved with current mobile display designs. The glass substrate with a build-in polariser for most commercial mobile display pixel CF layers is in the range of 0.25-0.3 mm. However, the ideal separation distance needs to be between 0.15 and 0.2 mm, because of the small sub-pixel width (20-30 µm) and its rectangular shape. We need to work with mobile display manufacturers to integrate the developed LC devices to the mobile display. Here we propose two optical configurations for the optimum auto-stereoscopic viewing experience, as shown in Fig. 18(a) and 8(b) respectively.

 

Fig. 18 The optimum auto-stereoscopic mobile display configurations. (a) Type 1: two glass substrates between the pixel CF layer and LC layer. (b) Type 2: a single glass substrate between the pixel CF layer and LC layer.

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The Type 1 configuration is to have two glass substrates between the pixel CF layer and LC lens layer. Glass substrate 1 of mobile displays in Fig. 18(a) needs to be 0.1 mm thick or less without the polariser film. It should be achievable with the latest development of ultra-thin glasses which the thickness ranges from 0.025 to 0.21 mm [24]. Glass substrate 2 of LC lens device also needs to be 0.1 mm thick in order to limit the separation distance to 0.2 mm. The polariser film, which is normally 0.1 mm thick, needs to be integrated to Glass substrate 3. The LC lens device is then glued to the display with an index matching glue of negligible thickness.

The Type 2 configuration is to have a single glass substrate between the pixel CF layer and LC lens layer, which means the LC lens device assembly needs to be integrated into the mobile display assembly process. Glass substrate 1 in Fig. 18(b) needs to be between 0.15 and 0.2 mm thick without the polariser film and it needs to have an ITO coating on one side. Type 2 configuration has the polariser film integrated to Glass substrate 3. Recently, a concept of active parallax barrier was demonstrated to reduce the viewing distance to below 300 mm, the work was based on a one-panel architecture with the help of a LC-on-polarising interlayer to reduce the separation distance [25].

For both configurations, the LC lens device can be switched on for auto-stereoscopic experience or switched off for normal 2D view. Each lenticular LC lens always covers two columns of sub-pixels, this is to ensure that there is a large enough distance ($d_x$) to produce a sufficient steering angle ($\alpha$) for the optimum viewing distance of mobile displays. The alternative off-axis LC lens that covers only one sub-pixel cannot produce the required steering angle.

5.2 Further improvements

The LC micro-lens device can be further improved in three aspects besides the proposed optimum configurations.

The first aspect is to reduce the crosstalk through two possible routes. The first route is to adjust, actually slightly reduce, the LC layer thickness so the LC lens EFL matches green wavelength better. With the current demonstrator, the LC lens EFL at the red wavelength matches the separation distance better and the crosstalk is lower for the red colour channel. Green, however, is perceived as the brightest of all three colour channels in human eyes and it is desirable to have the lowest crosstalk for green. The second route is to smoothen LC lens phase profile so it does not contain any discontinuities or non-steering region as shown in Fig. 13(b). The phase profile discontinuity as measured is mainly due to an artefact in the calculation of phase values from intensities. Nevertheless, better phase profiles can reduce the crosstalk, shown as the simulated values based on the ideal lens in Fig. 14. This can be achieved by using multi-electrode structures as described in [8,17] or an appropriate ratio between the LC lens half-width and its maximum phase depth. The correct ratio can generate a high quality LC micro-lens with a close-to-ideal lens profile [5]. For example, in our demonstrator the 8 µm LC layer generates an 8 π maximum phase depth at 550 nm, the LC lens half-width, determined by the sub-pixel size, is 35 µm. As a result, the measured phase profiles contain the discontinuity regions close to the LC lens boundaries with a ~5 µm width and the non-steering region at the centre of LC lens with a ~15 µm width as shown in Fig. 13(b). If the LC lens half-width is reduced to less than 25 µm i.e. the ratio of less than 3.1 µm/π, the LC lens phase profile will be much smoother. The smaller LC lens half-width is very much feasible because the latest smart phone displays have sub-pixel size between 20 and 30 µm, such as iPhone 6/7 series and Huawei P9/Mate 9 model. In addition, with the smaller LC lens half-width, the EFL will decrease to between 100 and 200 µm, which matches the separation distance in the proposed optimum configurations. The crosstalk can be further reduced if the mobile display’s viewing angle, determined by the scattering material in the CF layer, is reduced, but there is a dilemma of wide viewing angle 2D content or low crosstalk auto-stereoscopic experience. We think the mobile display is a personal device and sharing a small display with several viewers is not practical. Other works were done to reduce the crosstalk, such as using a double lens system, the first lens reduces the effective pixel dimension and the second lens works as the steering lens array to achieve an auto-stereoscopic effect [26,27].

The second aspect is to increase the LC lens switching speed for video contents. Currently the measured switching time from LC on to LC off for the device with an 8 µm thick LC layer is in the range of 80-100 ms when driven by 10 V rms. Because the switching time from LC off to LC on can be much faster in a few ms by overdriving [28], the switching frame rate of 10-12 Hz can be realized, which will deliver an auto-stereoscopic frame rate of 5-6 Hz in the two frames per image multiplexing mode. When we push the driving frequency to 20 Hz or even 30 Hz, the steering effect is compromised due to the slow LC relaxation from on to off. We expect that with the optimum configuration, i.e. a 5.5 µm thick LC layer and a 20-25 µm lens half-width with the LC lens EFL for green colour channel to match the separation distance (d= 0.15-0.2 mm), the switching time from LC on to LC off can reduce to 38-47 ms and the switching frame rate reaches 24 Hz for an auto-stereoscopic frame rate of 12 Hz. Further increase of the frame rate can be achieved by the development of new LC materials with high birefringence and fast switching speed, such as polymer dispersed/stabilized nematic LC and polymer-stabilized blue phase LC [29].

The third aspect is to improve the manufacturing process of the LC lenticular lens device. The fabricated full size LC device has intermittent electrodes and light beams from some columns of sub-pixels are not steered. This is caused by the chemical over-etching of long and thin electrode tracks, a dimension of 70,000×10 µm². Other electrode patterning processes, such as laser processing, can have a better control over the patterned dimensions, but these demonstrator devices were not laser processed because their size is larger than the patterning area of the laser system we have. In addition, the assembly with ultra-thin glasses of 0.15 mm is very challenging, the glass substrate breaks easily and LC layer uniformity is difficult to control mainly because the process was carried out manually without appropriate tools. We believe display manufacturers with the right equipment and process can have a much higher and more consistent yield.

6. Conclusion

We have designed, fabricated and demonstrated a full resolution auto-stereoscopic display based on the switchable LC micro-lens array on a full size mobile display. The large scale (4 inch) device aligns with the display panel perfectly at sub-pixel level and switches the entire display area uniformly with no distortion and no colour separation. Optical performance of the auto-stereoscopic display is characterised experimentally in terms of steering angle and crosstalk, and shorter separation distance between the mobile display pixel CF layer and LC lens layer with the closely matched LC lens EFL have demonstrated better performance with an improved viewing distance and lower crosstalk. Simulation models are set up to further optimise the performance, using the measured sub-pixel spatial light intensity distribution of the mobile display. The steering angle and crosstalk were simulated based on the ideal and measured lens profiles, and the results confirmed the advantages of the shorter separation distance. To further progress with the current study, optimum configurations have been proposed which need supports from display manufacturers to integrate the LC lens array assembly into the display panel fabrication. Areas that require further improvement have been identified and discussed, we believe the latest mobile display pixel size and tailored LC lens specifications would deliver a high quality, video rate auto-stereoscopic mobile display.