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Abstract
A method using a Fourier series is demonstrated to optimize an LED array for local dimming applications in liquid crystal display backlighting. The same optimization method is also suitable for LED displays in which the Moiré effect must be suppressed during photography with a minimum loss of spatial resolution. Initially, the angular intensity profile of a Lambertian LED is modelled when backlighting a Lambertian rear projection screen and compared to experimental data. An array of optimized LEDs and the resulting screen intensity pattern is then derived such that an intensity distribution with an intensity deviation of less than 2% is achieved. The angular intensity profile of the LED is modified using adjustable Fourier coefficients optimized according to an algorithm. The algorithm is designed to achieve an illuminated screen area of maximum size for a bounded LED backlight array to appear uniform in intensity to an observer. This Fourier series approach provides an elegant method to optimize the intensity profile of LED backlight arrays without the use of ray tracing.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
LEDs are currently used in a wide variety of display and lighting applications due to their low cost and high brightness. One major application of LEDs is within the backlight of a liquid crystal display (LCD). [1] The backlight often comprises an array of LEDs that, in combination with a diffusing sheet, is used to provide a light source with a uniform intensity output over the LCD area. This is known as full array backlighting as opposed to edge-lit LCD backlighting. The LCD dominates the display industry today due to its superior lifetime, low cost, high resolution and high peak brightness. [2] The achievement of uniform full array backlighting is dependent on parameters of the LED array including LED spacing, system depth and LED angular light distribution.
More demanding requirements for LCD backlighting also exist. [3,4] Local dimming is a method used to improve LCD contrast ratio, by only powering specific LEDs to illuminate selected areas of an image while leaving the rest of the LEDs unpowered. This allows for both very dark and very bright portions of an image and a substantial improvement in contrast ratio. It is a critical enabler of high dynamic range LCDs. Using local dimming to only power specific LEDs that are required for an image instead of the entire backlight array also improves the power efficiency of the LCD. A variety of local dimming strategies including edge-lit LCD backlighting and full array backlighting have been developed. [5–8] Local dimming using full array backlighting is well known to be a preferred solution.
LCD display technology is rapidly developing to satisfy High Dynamic Range (HDR) requirements. Mini-LEDs and micro LEDs are being studied for LCD backlighting due to their ability to provide many more local dimming zones. [9] Full array backlight local dimming-type LCD displays require the ability for portions of the LED array to be powered while adjacent LEDs are not powered. The halo-effect that is well known in local dimming LCDs can be minimized by ensuring minimum bloom or unwanted light spreading between powered and non powered portions within the LED backlight.[9] At the same time, the LED array should produce spatially uniform light over the desired portion of the LED array that is to be powered.
Assessments have been done on the power saving of local dimming compared to backlights that do not use local dimming. Optimizations to backlighting have been created based on the results of such assessments. One of these algorithms is described in [10] and includes application specific variables relevant to local area dimming. In contrast, the methods described in our work are more broadly applicable and rely on Fourier analysis as well as a novel algorithm to achieve more broadly applicable results.
Another key application for LED arrays in display technology that can benefit from our approach is in camera-ready LED displays. These LED displays, well known in the digital billboard industry, are increasingly used in TV studio and movie set applications. A challenge in these applications is the Moiré effect. During photography of an LED display, unwanted patterning appears in the camera image due to the grid of camera pixels interacting with the grid of LED pixels. [11] The current solution to mitigate the Moiré effect is to place a diffusing screen in front of the LED display to produce more uniform light emission over the LED display surface; however this results in a drop in effective display resolution. A method that would mitigate the Moiré effect while also causing minimal degradation of the resolution of the image would be valuable for camera-ready LED displays. Once again, a diffused LED array capable of substantial uniform light emission when a region of one or more LEDs is powered while an adjacent region of LEDs is not powered is required, such that minimum unwanted light spreading occurs. Excessive light spreading limits effective display resolution.
The spatial extent of light spreading that occurs in either a local dimming-type LCD backlight or in a camera-ready LED display depends on the degree of diffusion of the diffusing sheet placed in front of the LCD array. With insufficient diffusion, it is not possible to produce acceptably uniform illumination for the application at hand. If diffusion is increased, then spatial resolution decreases and blooming increases.
Increasing the spatial density of LEDs will lead to an improvement in spatial light uniformity and a reduction in light spreading since a lower degree of diffusion is then required, however cost constraints limit the extent to which this approach may be applied. There is, however, another way to optimize the LED array to minimize the requirement for diffusion: The angular emission profile of each LED within the LED array can be optimized to allow for a diffuser having a minimum degree of diffusion to enable substantially uniform light emission. This will then lead to a reduction in light spreading or bloom.
A common way to alter LED angular emission profiles is to place lenses over the LEDs. Many lens types are known. Freeform lenses, multiple curvature lenses, and double freeform lenses are well developed and may be manufactured cost effectively using plastic molding.[12–14] Methods to design freeform lenses include the differential equation method, the tailoring method, the simultaneous multiple surfaces method (SMS), the parameter optimization method, and supporting paraboloids method. [15–17]
The purpose of this paper is to investigate details of modified LED angular emission profiles, to study the impact of these profiles on light emission uniformity from an array of such modified LEDs, and more specifically to minimize light spreading or bloom. For the first time, an approach has been undertaken in which modified LED angular profiles are modeled using Fourier series in conjunction with a minimizing algorithm. This is shown to be a powerful tool to optimize LED array design and to minimize the degree of diffusion required.
Modelling and optimizing backlight uniformity commonly includes the Liquid Crystal layer as well as the backlight and the diffusing layer. [18] The method presented in this paper is a more general model that will be applied to a basic setup of an LED backlight and a Lambertian screen in order to display our method in the most simplistic way. The method models an area of intensity on a screen from an LED array using a Lambertian intensity distribution of an LED. After modeling the Lambertian distribution using a Fourier series, the model is extended to allow for modification of the intensity distribution, and its application to all LEDs in the array such as producing a substantially uniform intensity distribution with a local deviation of less than 2%.
In order to control bloom, a minimization algorithm minimizes light extending beyond a boundary established around a zone of illuminated LEDs. The method introduced in this article aims to improve local dimming. Implementing this method could also increase the distance between LEDs, thus reducing cost while still maintaining uniformity.
Light intensity angular emission profiles resulting from this model could subsequently be implemented using commercially available LEDs combined with low cost molded plastic lenses. In this paper, conventional LEDs spaced 5cm apart will be studied to illustrate the modelling and optimization methods. It is trivial to change the scale to suit smaller LED pitches.
2. Theory and methods
2.1 Introduction
When an array of LEDs illuminates a rear projection screen, an observer of the screen sees an image that depends on the distance between the screen and the LED array, the spatial density of LEDs, and the angular intensity profile of the light from each LED. When considering local dimming or moiré elimination, the spread of light from each LED on the screen needs to be as limited as possible.
LED arrays may result in non-uniform intensity patterns on the rear projection screen when the LED array is positioned too close to the screen and/or the spatial density of the LED array is low. Human visual perception is a key component in determining uniformity. If luminance intensity deviations within luminous intensity patterns are less than approximately 2%, then the illuminated area will appear substantially uniform in intensity to the average observer. [19]
Metrics to assess the illumination uniformity based on the human visual system perception have been developed. [20] Sparrow’s Criterion is described as the resolution limit when attempting to resolve the joint intensity map of two separate disks that are close together but have equal intensity. In [21] a polynomial equation was used to represent intensity of an LED array and Sparrows Criterion was used to achieve uniform illuminance in the central region, while a second and third method were used for the non central regions. In [22] Sparrow’s Criterion was compared with a contrast sensitivity function (CSF) as a way to evaluate an LED array lighting system. CSF is a way to describe how well the human visual system is able to detect patterns. Due to the nature of Sparrow’s Criterion, which does not include the human visual system, authors of [22] chose to use CSF as the criterion to create a uniform lighting condition for LED arrays.
An example of strong LED illumination patterns on a screen can be seen in Fig. 1.
Fig. 1. LED array pattern on a screen due to an LED backlight located 3 cm behind the screen. LED pitch is 4.96 cm and each LED is a Lambertian emitter.
2.2 LED intensity profiles
Firstly, the intensity profile of an LED must be considered. An LED intensity profile is represented graphically as intensity versus angle $\theta $ between the direction of emitted light and the LED optical axis. For a lambertian LED, the luminous intensity varies as the cosine of this angle. In general most LEDs will have a Lambertian radiation pattern. This LED intensity is represented by Eq. (1).
The intensity profile of a substantially Lambertian light source is depicted graphically in Fig. 2.Fig. 2. Intensity profile of a Lambertian source, using both Polar coordinates (Left), and Cartesian coordinates (Right).
2.3 Screen interaction
An ideal rear projection screen would normally be Lambertian in nature. The screen situated in front of an LED array, if Lambertian, should have ideal diffuse transmittance, in that it takes all incoming light from the LEDs and transmits the light through the screen and diffuses it equally in all directions. The behavior of a Lambertian screen is depicted in Fig. 3A. A non Lambertian screen would have directional behaviour as shown in Fig. 3B. It is assumed that the rear projection screen is Lambertian.
Fig. 3. Lambertian surface A: Ideal Diffuse transmittance, B: transmittance with directional component (non Lambertian)
Assuming a single Lambertian LED illuminates a Lambertian rear projection screen, the screen light intensity as a function of $\theta $ as shown in Fig. 4 will be described by Eq. (2) and Fig. 5.
Justification for the fourth power or ${\cos ^4}\theta $ dependence in Eq. (2) will now be reviewed. The first cosine term comes from the LED, which is shown in Eq. (1).Fig. 5. Expected screen intensity versus angle θ from Eq. (2). Both LED and screen are assumed Lambertian, and θ is defined in Fig. 3.
Two cosine terms arise from the increasing distance between the LED and the screen as a function of angle. Brightness is inversely proportional to the square of the distance as shown in Fig. 6.
Since $\frac{D}{H} = \cos \theta $, therefore $I = {I_0}{\cos ^2}\theta $.The fourth cosine term arises from the amount of light that appears on the screen due to the surface area subtended by a differential angular range of $d\theta $. As depicted in Fig. 7 dx shows the size of the spot where light will arrive and this is increased as the magnitude of the angle θ gets larger.
The intensity output of an LED on a screen will initially be analyzed for screen output described by Eq. (2). This will be considered a baseline for modifications.2.4 Factors that affect image
The modelling depicted in this paper is based on a square LED array having 5 cm LED pitch. The chosen goal in this paper is to achieve optimum results with a height of 3 cm between the screen and the LED array since these values match to a physical LED array that we used to validate our modelling. These values may be scaled as desired to suit other geometries. Modelling will be restricted to angular emission profiles that are symmetrical about the optical axis of a given LED.
Examining two LEDs placed side by side with the LED spacing and screen height mentioned above, the resulting intensity profile can be seen in Fig. 8. The resulting percentage deviation is approximately 35.8% which is much higher than 2% and therefore two bright spots would be perceived on the screen. The relationship between screen height and percentage deviation for multiple LED spacing options is analyzed in Fig. 9. It can be seen for an LED spacing of 5 cm the screen height would need to be 5 cm to produce a percent deviation of 2% or less when only 2 LEDs are considered. It is clear that for LED spacing of 5 cm and screen height of 3 cm as an example, the intensity profile of the LED must be modified to achieve a uniform screen appearance.
Fig. 8. Overlap of 2 adjacent LEDs and the resulting intensity profile (green) that an observer would see on a screen due to 2 LEDs. The difference between the two peaks and the valley between them is 34.28% of the intensity of the peak. x is defined in Fig. 7
Fig. 9. Spatial percent deviation in luminous intensity as a function of screen height for various spacing between two adjacent LEDs
2.5 Fourier representation
The angular intensity profile of an LED will now be represented using a Fourier series.
A second optimization requirement is now introduced. Whereas uniform screen illumination may be achieved using very wide angle lensed LEDs, our goal is also to minimize bloom. To this end, we invoke the use of a finite or bounded LED array. This finite LED array will result in limited spread of light beyond the bounded array. The Fourier representation of Eq. (7) which has a limited number of terms cannot properly represent wide angle illumination commonly found in low cost full array backlit LCD displays. Since wide angle illumination is not wanted, the limited number of terms is acceptable and it allows us to effectively optimize small angle luminance dependence.
A constraint is therefore introduced by means of a square “window of uniformity”. This window is defined by the area on the screen illuminated by the finite LED array providing screen intensity deviation less than +/- 1%. Our algorithm seeks to maximize the size of the window.
This window of uniformity is optimized to be as large as possible and thereby ideally close to the size of the bounded LED array. The window can only be optimized by changing the Fourier coefficients. An example of a finite LED array and a resulting window of illumination are shown in Fig. 10
Fig. 10. Window uniformity is defined by screen area over which uniformity may be achieved (red) in this example overlaying an 8 × 8 LED array. Outside of the window, screen illumination will drop off rapidly and deviates by more than +/- 1%.
Matlab code has been created to optimize the Fourier coefficients in Eq. (7) to maximize the window of uniformity Given an estimate of the starting coefficients for the Fourier expression, the code will then create an array and attempt to optimize the window according to the parameters by adjusting the coefficients.
Two parameters are used for seeking an optimized condition. The first parameter is the deviation parameter defined by the percentage difference between the maximum and minimum intensities within the window, representing the flatness of the screen intensity profile. A square window is created based on the area that respects this threshold value. The second parameter is the side length of the square window which we seek to maximize. Eq. (8) defines R, a merit function, to be the ratio between the deviation parameter and the window size. R is to be minimized in order to find a maximum window size while keeping an appropriately small deviation parameter within the threshold.
The merit function does not explicitly include a variable to address bloom, but rather, by finding a balance between intensity uniformity and area size, bloom is inherently addressed. This occurs because, by seeking to maximize the area of uniformity from a finite LED array size, localized LED levelling is optimized and the spatial extent of bloom is thereby automatically minimized.Matlab code now uses the minimization of R to find the coefficients in Eq. (7) by a successive approximation algorithm.
3. Results
3.1 Experimental results
A Cree LED (Xlamp XQ-E) in conjunction with an almost Lambertian screen (Stewart Filmscreen Aeroview 70) were used to test and validate modelling. The screen was situated 3 cm above the LED. The spot intensity at the screen was measured with a Minolta photometer while the LED was moved in 1 dimension so as to acquire light from limited LED angles between −76.2 and 37.1 degrees. The angular limitation is due to physical limitations of the movement system. The intensity profile in Fig. 11 was obtained. In order to further validate these results, a Teledyne Dalsa Genie TS camera was used to obtain a luminance distribution of the screen for all angles. This is also presented in Fig. 11. Good agreement between measured and expected Lambertian profiles is clear suggesting a Lambertian model is sufficient to represent an experimental system. Other screens were also tested, but they did not have a sufficiently Lambertian response and were therefore not used. Non Lambertian screens introduce viewing angle effects and are therefore not suitable unless additional viewing angle considerations are modelled, which go beyond the scope of this paper.
Fig. 11. Intensity profile of XQ-E high Intensity LED on a screen; Photometer results (green), Theory of Eq. (2) (blue), Camera results (red)
Using these LEDs and the Aeroview 70 screen, an LED backlight array situated behind the screen was constructed and analysed. The LED array was 3 cm behind the screen and the LED array had pitch of 5 cm. The resulting luminance pattern is shown in Fig. 12A. The LED pattern is quite discernible. A model based on the single LED intensity profile of Eq. (2) is shown in Fig. 12B which comprises a superposition of luminance profiles of every LED in the array.
3.2 Fourier coefficients
In Section 2.5 a model was developed that would find a set of coefficients that, when applied to Eq. (7), would create an intensity profile for one LED that would be optimized to create a uniform intensity profile on a screen for a finite LED array. The LED array used to model results is a 20 × 20 array with pitch of 5 cm and a distance of 3 cm between the LED array and the screen. Inserting these parameters into the model, the optimized coefficients found via the successive approximation algorithm are shown in Table 1. These coefficients, when applied to Eq. (7), provide the ideal intensity equation for each individual LED. These values were chosen to model an experimental setup that exists in our lab, however the dimensions and results may readily be scaled
Table 1. Fourier coefficients determined via Matlab algorithm.
3.3 Optimized intensity profile
Using the coefficients found in section 3.2 and applying them to Eq. (7), the optimized LED angular luminance profile is displayed in Fig. 13. Comparing the original Lambertian distribution(blue) to the new optimized function(red), it can be seen that the angular spread of light has been increased for smaller angles, while remaining similar to the original profile at larger angles.
Fig. 13. Lambertian Intensity profile of cos4θ (blue) compared to the new optimized intensity profile (red)
An LED array in which each LED is modified to have this optimized angular intensity profile is now simulated. In Fig. 14 a full 20 × 20 array is modeled with the original Lambertian distribution. In Fig. 15 the same 20 × 20 array is modeled with the new distribution. Clearly, the modified angular distribution results in a substantial improvement in uniformity.
3.4 Analysis of uniformity and degree of edge effects
The results displayed in the previous section appear very promising to the naked eye, but they must be analyzed further. The percentage deviation that is being aimed for is +/−1%, over as much of the area of the array as possible. A topographical map of the percent deviation of the intensity was created and is shown in Fig. 16.
Figure 16 shows details of areas where the percent deviation in luminance increases above the threshold limit of 2% of the max intensity. The edge of the window of uniformity can be seen at locations in x and y from 10–80 cm. There are some small regions within this window, where the percent deviation slightly rises above 2%, however it does not reach more than 2.1%. The asymmetry is due to a propagation of a small rounding error.
The size of this window of uniformity needs to be compared to the size of the array. A visual aid is presented in Fig. 17 by superimposing both the LED array and the window. There is a difference between the window of uniformity (yellow) and the edge of the LED array (pink). The difference between these two edges is approximately 10 cm in width on every side. Based on the spacing between LEDs used in this model, the area between the edge of the LED array and the window of uniformity would correspond to 2 extra rows/columns of LEDs. Knowing this, if the size of the required display is known, there would need to be a buffer of 2 rows/columns of LEDs around the display area to allow for edge effects that the viewer would be exposed to. This effect could be scaled depending on the spacing of the LEDs.
Analysing this edge area, the percent deviation increases to 5% halfway between the edge of the window of uniformity and the edge of the LED array, and it reaches a maximum of approximately 15% at the edge of the array. Human vision is known to disregard small deviations near a boundary and therefore this increase in deviation would not be very noticeable to the average observer. Edge effects comprise a complex area of study for human vision detection.
3.5 Analyzing bloom
Figure 18 describes how the overall shape of the intensity profile of a 20 × 20 LED array changes as the R parameter described in Section 2.5 is increased. As R is increased a more prominent halo effect is displayed as well as a decrease in uniformity.
Fig. 18. Visually analyzing how increasing R affects bloom. The first row depicts a cross section of the intensity profile of a 20 × 20 LED array. The bottom row shows a top down view as an observer would see on a Lambertian screen.
When R = 0.03 the window of uniformity is approximately 70cm2. For R = 0.14 the window of uniformity is approximately 14cm2 both with an intensity deviation of 2%. For R = 0.2 the intensity deviation never reached 2% but instead resulted in a window size of 20cm2 for a deviation of 4%. In order to change R, the coefficients were modified, producing a new window size where the percent deviation was minimum and constant.
R does not specifically have a bloom parameter and therefore there is no direct connection between bloom and R. The merit function R attempts to maximize uniformity and also to maximize the size of a substantially uniform area. However, based on the visual results of Fig. 18 there is a correlation between R and bloom, as was predicted in Section 2.5.
4. Discussion
A substantially uniform intensity profile has been modeled using an optimized intensity distribution that must be applied to all LEDs in an array. In order to reduce this approach to practice the required intensity profile must be achieved using optics applied to each LED. A lens or other optics would be designed that provides the intensity profile discovered in Fig. 13 and represented by Eq. (7). Applying one lens per LED in an array would allow for a uniform array equal to that of the model presented in Section 3.
This method provides optimized illumination uniformity, and also optimizes local dimming. Because this method does not require a broad expanse and overlap of light from the LEDs, turning off particular LEDs would continue to allow for substantially uniform intensity where the remaining LEDs are powered. As introduced previously in Section 3.4, the light spread from each LED is small, allowing for accurate local dimming and minimizing bloom. This method of designing a uniform array would also minimize the moiré effect while minimizing unnecessary bloom and a corresponding loss of display resolution in camera-ready LED displays.
The use of a Fourier series to represent the Lambertian distributions of an LED and screen represent a powerful approach to optimizing the LED angular intensity profile. The superposition of light from a resulting LED array may be used to achieve optimum properties of the individual LED.
Comparing the results of this Fourier method to that of the polynomial method described previously in [21], the optimized intensity profile has a wider peak centered at an emitting angle of 0° and dropping off at +/− 80°. The polynomial method resulted in peak light emission at 60°–70° off axis with less light emitted at smaller angles. Each method provided drastically different shapes while providing high uniformity, yet the polynomial method did not address the issue of bloom, and would introduce substantial bloom compared to our results.
The model shows this Fourier method to be very promising. Work on lens design to implement the Fourier series modelling in this work is being pursued and future publication is planned.
This method could be extended to non Lambertian LEDs, as the Fourier series is powerful. Due to the nature of most LEDs being all or nearly Lambertian, only a Lambertian LED was modeled in this paper. Attention was restricted to a Lambertian screen due to the assumption of zero viewing angle effects. Viewing angle independence was maintained.
5. Conclusion
A new method for modelling and creating a uniform intensity distribution for a rear projection screen illuminated by an LED array has been proposed and demonstrated. The method is a very general approach that provides an elegant solution without the use of ray tracing. The Lambertian intensity distribution of an LED has been modeled and expressed using a Fourier series. The coefficients of this Fourier series expression were then modified using an algorithm that optimized both the spatial uniformity of the screen intensity and the screen area, described as a window of uniformity, over which this uniformity corresponded to less than a +/−1% luminance deviation. The optimized Fourier coefficients may be applied to individual LEDs in an array to create a window of uniformity that approaches the area of the LED array. The resulting window of uniformity was further analyzed and found to retain adequate uniformity, and to be sufficient in size to allow for improved local dimming. Due to the large spacing between LEDs, this method is shown to lead to a low cost approach to LCD backlighting, as well as an effective approach for LED displays where photography is required.
Funding
Mitacs.
Acknowledgements
Thanks are given to Siwei Ma and Sam Peters for many useful discussions, as well as Tim Van Boxtel for Matlab assistance.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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