1. Introduction

Nowadays, light emitting diode (LED) is rapidly taking the place of traditional light source in multiple application fields due to its advantages such as high luminous efficiency, long work life, compact volume, no use of mercury and so on [1]. Among these applications, planar uniform lighting is a conventional requirement in LCD backlight, panel light, projector light etc [2–8]. The most important and convenient way to achieve planar uniform lighting is to use LED array lighting system (LALS) [2–8], in which uniform lighting on a target plane is produced by an LED array placed beneath, and then the illuminated target plane is observed by human vision system in transmission or reflection. Taking direct LED backlight for liquid crystal display (LCD) shown in Fig. 1 as an example, a thin diffuse sheet is placed upon an LED array with the most conventional foursquare arrangement to prevent human vision system from seeing the LEDs directly, and then uniform lighting transmitting through the thin diffuse sheet is desired on the target plane. Structural parameters including system thickness z, LED pitch d, and LED’s spatial radiation distribution I(θ,φ) (θ and φ is zenith and azimuth view angle of the LED respectively), as well as the viewing condition of human vision system including viewing distance H and viewing angle α are all shown in Fig. 1. Region around the obsevation point in the center of the target plane is enlarged in the right of Fig. 1 with a rectangular coordinate system. The origin is surrounded by four LEDs.

 

Fig. 1 LED array lighting system observed by human vision system with foursquare arrangement and an enlarged view of the region around the observation point in the center

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Generally, brightness perceived by human vision system is proportionate to illuminance value on the target plane under normal condition i.e. design of uniform lighting is usually equivalent to design of uniform illuminance distribution [2–8]. For an LALS large enough, illuminance distribution is regular and periodic in most region on the target plane except the edges. In this paper, our study is focused on the main and normal lighting region in the center rather than the edges, and then illuminance distribution function E(x,y) around the observation point can be expressed in Eq. (1)

In Eq. (1), E₀ is illuminance of the original point, accordingly, $\overline{E}(x,y)$ is normalized illuminance distribution function; $E_i(x,y)$ is illuminance distribution produced by each LED and M is the number of LEDs. Since the thin diffuse sheet eliminates the pattern of LEDs but hardly changes the illuminance distribution, it can be ignored in the calculation of illuminance distribution, as former research did [2, 3].

While designing an LALS, uniform lighting is usually expected to be achieved with small system thickness and large LED pitch in consideration of practicability, cost, appearance etc [2–5]. Empirically, smaller system thickness or larger LED pitch may lead to worse uniformity [2–5]. However, it is neither practicable nor low-cost to experiment or simulate repeatedly to determine the optimum structural parameters. In this case, a critical criterion for uniform lighting, or uniform illuminance distribution, is required to construct a quantitative uniformity condition, from which optimum system thickness or LED pitch can be calculated.

Former research developed by Moreno et al has proposed critical uniformity condition for LALS based on Sparrow's Criterion (SC) [2, 7]. In SC method, illuminance distribution is judged to be “flat” on a point when second derivative of the distribution function equals to zero here. However, SC is used for the resolution of two light patterns by judging whether there is some central “dip” in the pattern combined by them [9], human vision effect is not considered here whereas most LALS are observed by human vision system.

In this paper, contrast sensitivity function (CSF), which describes the ability of human vision system to perceive a grating pattern, was novelly adopted as a critical criterion [8, 10–18]. Through CSF method, design parameters including system thickness, LED pitch, LED’s spatial radiation distribution and viewing condition can be analytically combined. To give an example, optimum system thicknesses and LED pitches were calculated and compared with those got through SC method in a specific LALS with foursquare arrangement, given viewing condition and different types of LED. Results calculated through CSF method were demonstrated to be more “critical” than those calculated through SC method if human vision effect is considered. Additionally, multiple critical values of structural parameters and uniformity varying with structural parameters non-monotonically were found in LALS with non-Lambertian LED while that with Lambertian LEDs does not exhibit so. This abnormal phenomenon was analyzed, based on which, a practical design method of LALS was proposed.

2. Introduction to SC method

It is necessary to introduce SC method before CSF method is discussed to give a comparison. In this section, an LALS was specified and algorithmic method for optimum system thickness z or LED pitch d was proposed.

The specified LALS, which will be studied several times in this paper, was set to include an N by N Lambertian LED array with foursquare arrangement, which is denoted as LALS-Lambertian. Accordingly, I(θ,φ) equals to I₀cosθ where I₀ is the central luminous intensity corresponding to θ = 0° and illuminance distribution function E(x,y) can be expressed in Eq. (2)

where r_ij is distance from each LED to point (x,y).

If N is set to be 10 with consideration of both accuracy and calculability, normalized illuminance distribution function $\overline{E}(x,y)$ in Eq. (2) can be given in Eq. (3)

Critical uniformity condition on point (x,y) in direction l is ${\partial }^2\overline{E}/\partial l^2=0$ according to SC [2–5, 9]. For LALS-Lambertian, planar uniformity can be usually judged by uniformity in x-axis (or y-axis) direction, as${\partial }^2\overline{E}/\partial x^2=0$, at several points. Taking both symmetry and accuracy into consideration, 6 “characteristic points” in one eighth of a grid, as P₁-P₆, were selected for calculation, as shown in Fig. 2 . And then system thickness z and LED pitch d can be derived from each other through the calculations on these 6 points.

 

Fig. 2 Six characteristic points used for critical uniformity condition calculation with SC method for LALS-Lambertian.

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If LED pitch d is set to be 10mm, 11mm, 12mm…20mm respectively, critical values of z corresponding to P₁ to P₆ were calculated (critical z was determined to be the value that can lead to the minimum of ${\partial }^2\overline{E}/\partial x^2$ if no root can be found). Table 1 shows the results and the maximal ones marked with “*” among every 6 values were empirically determined to be the optimum system thicknesses to guarantee uniform distribution at all 6 points. It can be seen that: (1) critical values of z in the 6 rows of Table. 1 are all perfectly linear with respect to LED pitch d; (2) P₃ vertically above LEDs was found to be the most “difficult” to be uniform than other points, which is in accordance with design experience.

Table 1. Critical values of system thickness corresponding to 6 characteristic points under given LED pitches

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On the other hand, former research [2, 3] tended to only select the center point on the target plane to calculate the root of ${\partial }^2\overline{E}/\partial x^2=0$. Specifically, if N is even, the single point is P₆ surrounded by LEDs and if N is odd, the single point is P₃ just above an LED. It can be revealed fro1m our study above that the selection of single point in former research is appropriate when N is odd; however, when N is even, such a selection will cause a little inaccuracy because calculation at P₃ can bring about a larger critical z than that got at P₆ whereas the selection of single point cannot take P₃ into consideration.

Similar calculations were implemented from given z to optimum d and calculation results are shown in Table 2 . Similar phenomena were found that critical z is linear with respect to d and P₃ is the most “difficult” point to be uniform.

Table 2. Critical values of LED pitch corresponding to 6 characteristic points under given system thicknesses

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SC method can provide reasonable reference while LALS is to be designed. Once a specific type of LED is utilized, optimum value of system thickness can be calculated after LED pitch is given, and vice-versa. Even, a reversing design from structural parameters to LED’s SRD can be implemented [3, 5]. However, LALS is usually observed by human vision system and SC method is not able to take human vision effect into consideration. So a new critical criterion, as contrast sensitivity function, will be adopted in the following study.

3. Application of CSF method in LALS-Lambertian

Contrast sensitivity function (CSF) focuses on the contrast sensitivity (CS), as the reciprocal of contrast ratio, of a sinusoidal/cosinoidal luminance modulated grating. As mentioned before, luminance can be proportionally reflected by illuminance under normal viewing condition, so illuminance distribution of the grating is used to calculate CS [8], as given in Eq. (4).

where L_max and L_min is the maximal and minimal luminance in a single cycle of a sinusoidal/cosinoidal luminance modulated grating; E_max and E_min is the maximal and minimal illuminance in this single cycle.

The theory of CSF, which has been developed for decades of years by optical and ophthalmological researchers [10–18], states that threshold CS, as denoted as CS_th, for human vision system to distinguish a sinusoidal/cosinoidal luminance modulated grating from uniform lighting relies on spatial frequency ω in cycle/degree under a given viewing condition, namely CS_th is a function of ω. The function expression of CS_th(ω) depends on many factors including observer’s age, gender, health condition, ambient brightness and so on [13–15, 18]. As this paper is focused on the application of CSF as a critical criterion for LALS with uniform lighting, but not how to construct an accurate CS_th(ω), the popular and typical Barten model [13–15], as given in Eq. (5), is adopted for the following analysis.

where a = 540(1 + 0.7/L)^-0.2/[1 + 12/X/(1 + ω/3)²], b = 0.3(1 + 100/L)^0.15 and c = 0.06. L is ambient brightness in cd/m² and X is the angular pattern size in degrees, which is calculated from the square root of the pattern area.

X is set to be 45 degree, which is conventional for LALS application like backlight for LCD; L is set to be 500cd/m², which is also a conventional level in illumination applications. With these assumptions, the functional image of CS_th(ω) can be plot in Fig. 3 , from which we can see a band-pass characteristic of human vision system. Obviously, region above or below the curve in Fig. 3 indicates uniform or nonuniform lighting respectively.

 

Fig. 3 Functional image of contrast sensitivity function (CSF) where region above or below the curve indicates uniform or nonuniform lighting respectively

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In the LALS-Lambertian studied before, around the observation point, illuminance distribution in x-axis (or y-axis) direction is periodic with a cycle equal to LED pitch d, therefore it can be regarded as an illuminance modulated grating to investigate with CSF. In the study with SC method, 6 characteristic points were selected to cover one eighth of an LED grid. Similarly, in consideration of symmetry, several lines parallel to x-axis, which are called “characteristic lines”, were also selected to cover half of an LED grid to make the study with CSF method reasonable. As shown in Fig. 4 , Q + 1 characteristic lines in half of an LED grid were selected and illustrated. They were uniformly distributed on y-axis with y coordinate y_k = kd/2Q where k = 0,1,2…Q respectively. And then, these normalized one dimensional illuminance distributions $\overline{E}(x,y_k)$, which are obtained by substituting y by y_k in $\overline{E}(x,y)$, become our study objects. Uniformity of all the$\overline{E}(x,y_k)$, which are judged with CSF, can result in a planar uniform lighting.

 

Fig. 4 Characteristic lines used for critical uniformity condition calculation with CSF method

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Additionally, spatial frequency ω can be regarded as a constant around the observation point, as given in Eq. (6)

The last problem to construct critical uniformity condition with the aid of CSF is that normalized one dimensional illuminance distribution function$\overline{E}(x,y_k)$ is periodic but not sinusoidal or cosinoidal around the observation point, which causes that Eq. (4) cannot be directly used to calculate CS. Fortunately, the way CSF is applied in perceiving square wave has been discussed by former researchers [16, 17] and is introduced in the appendix section at the end of the paper, which can be used as a reference.

Similar to the way CSF is applied in perceiving square wave, Fourier series was used to decompose $\overline{E}(x,y_k)$ into sum of cosine components with different orders. As waveform of $\overline{E}(x,y)$ is much more gentle than that of square wave, amplitude of higher order harmonics will be much smaller than that of the fundamental’s. Hence the impact of higher order harmonics can be ignored while judging the uniformity of $\overline{E}(x,y_k)$ and then $\overline{E}(x,y_k)$ is accurately regarded as the sum of mean and fundamental harmonic, which is a cosinoidal illuminance modulated distribution. Equation (4) can be now used to calculate CS of $\overline{E}(x,y_k)$ and CS is just obtained as the ratio of the mean to the amplitude of fundamental harmonic.

For instance, in LALS-Lambertian with d = 10mm, if system thickness z is set to be 5mm and 10mm, Fourier series of $\overline{E}(x,y_0)$ can be calculated and normalized, as given in Eq. (7). It can be seen that amplitudes of secondary and higher order harmonics are smaller than that of the fundamental harmonic by several orders of magnitude, so it is extremely accurate to use amplitude of the fundamental harmonic to calculate CS for the original illuminance distribution.

Based on the work above, system thickness z, LED pitch d, viewing distance H and viewing angle α for an LALS-Lambertian with critically uniform lighitng can be analytically combined through CSF.

Then viewing condition were determined to be H = 500mm, α = 0° and average luminance was assume to be maintained at a level of 500cd/m². Optimum system thickness z was to be solved under d = 10mm, 11mm, 12mm…20mm, which is the same problem that has been already solved with SC method. For calculability and accuracy, Q was set to be 5.

When d = 10mm, CS of $\overline{E}(x,y_k)$corresponding to six characteristic lines were calculated as the ratio of the mean to the amplitude of fundamental harmonic under z from 1mm to 15mm (not from 0 to ensure appropriate physical significance). CS_th varying with z from 1mm to 15mm was also calculated (In fact CS_th is only dependent on LED pitch d and is a constant with respect to z here). Then differences between CS of $\overline{E}(x,y_k)$ corresponding to six characteristic lines and their thresholds CS_th were calculated. Figure 5 gives the variation behavior of these differences with z from 1mm to 15mm. And then single critical z was found: when z equals to 12.43mm, CS of all the $\overline{E}(x,y_k)$ are larger than their thresholds, in other words, all the lines in Fig. 5 are upon the zero line in “uniform region”, which can lead to critical uniform lighting. Similar phenomenon was found that region vertically above LEDs corresponding to y₅ is the most “difficult” to be uniform. With the same method, optimum system thicknesses for other given LED pitches were calculated, as shown in Table 3 .

 

Fig. 5 Differences between CS of $\overline{E}(x,y_k)$ corresponding to each characteristic line and their thresholds under different system thicknesses z in LALS- Lambertian with d = 10mm. Region above or below the zero line indicates uniform or nonuniform lighting respectively

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Table 3. Optimum system thicknesses for given LED pitches in LALS-Lambertian calculated through CSF method

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Then system thicknesses z = 10mm was given. CS of $\overline{E}(x,y_k)$corresponding to each characteristic line were calculated under d from 1mm to 15mm. CS_th varying with d was also calculated by Eq. (5) and Eq. (6). Then the differences between them varying with d are shown in Fig. 6 . It can be seen that the curves are not monotonic, which is explained below: In Fig. 5, d was given, thus CS_th is constant while z is changing. Since CS of all the curves increase with z, the differences between them also increase with z, resulting in a monotonic curve. On the other hand, in Fig. 6, CS of all the curves decrease with d; at the same time CS_th also varies with d. Referring to Fig. 3 and mapping d to ω with Eq. (6), CS_th will increase and then drop down while d increases from 1mm to 15mm . So the difference between CS and CS_th will decrease, increase, and then decrease to be smaller than zero orderly, as reflected by Fig. 6. Noticeably, when d is around 2mm, which corresponds to the best perception of human eyes, such a small ratio of LED pitch to system thickness can even lead to a nonuniform distribution. This is abnormal to former research based on SC [2, 3, 5]. However, distribution corresponding to d≈2mm is just below the threshold slightly, which may be interfered by the inaccuracy of adopted CSF model or assumed viewing conditions. More importantly, larger LED pitch is always desired, so d = 7.87mm was finally determined to be the optimum one.

 

Fig. 6 Differences between CS of $\overline{E}(x,y_k)$ corresponding to each characteristic line and their thresholds under different LED pitches d in LALS-Lambertian with z = 10mm. Region above or below the zero line indicates uniform or nonuniform lighting respectively.

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With the same method, optimum LED pitches for given thicknesses 11mm, 12mm…20mm were also calculated and shown in Table 4 .

Table 4. Optimum LED pitches for given system thicknesses in LALS-Lambertian calculated through CSF method

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Now a series of optimum system thicknesses and LED pitches were calculated through SC and CSF method respectively. For comparison, these results were plot in Fig. 7 , from which we can see that: (1) smaller optimum system thicknesses and larger optimum LED pitches were achieved through CSF method i.e. more relaxed uniformity condition and better design parameters that can lead to better practicability, lower cost and more attractive appearance were produced by CSF method in comparison with SC method; (2) optimum design parameters achieved with SC method are perfectly linear with respect to given parameters, which can be verified by the data in Table 1 and Table 2. However, by calculating the data in Table 3 and Table 4, a little nonlinearity of optimum parameters achieved with CSF method with respect to given parameters was found, which can be attributed to that CS_th varies with LED pitch d. As verification, neither red line in Fig. 7 is perfectly straight line.

 

Fig. 7 Optimum system thicknesses and LED pitches achieved through SC and CSF method. Blue or red line denotes SC or CSF method respectively: (a) from given LED pitches to optimum system thicknesses; (b) from given system thicknesses to optimum LED pitches

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To verify whether uniformity condition achieved through CSF method is trustworthy, several visual illuminance distributions generated by MATLAB software corresponding to optimum design parameters achieved through SC and CSF method were shown in Fig. 8 and Fig. 9 . As optimum parameters were calculated for certain viewing condition (H = 500mm, α = 0°), observer should be adjusted to reproduce the predetermined viewing condition. Each sub figure in Fig. 8 and Fig. 9 horizontally corresponds to a length of 14d and covers 2.77cm on the computer display in 100% view mode while original spatial frequency can be calculated through Eq. (6). So one should make his eyes 9.88cm, 4.94cm, 12.45cm, 16.62cm, 5.78cm and 8.31cm vertically above center of these figures while observing Fig. 8(a), Fig. 8(b), Fig. 9(a) (left), Fig. 9(a) (right), Fig. 9(b) (left) and Fig. 9(b) (right).

 

Fig. 8 Comparison of illuminance distributions corresponding to optimum z achieved through CSF method and SC method: (a) under given d = 10mm; (b) under given d = 20mm

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Fig. 9 Comparison of illuminance distributions corresponding to optimum d achieved through CSF method and SC method: (a) under given z = 10mm; (b) under given z = 20mm

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Then six students in our group were involved in a simple visual experiment to judge whether these illuminance distributions in Fig. 8 and Fig. 9 were uniform. Considering that different spatial frequency may lead to different CS_th, observers were also moved from the initial observation position towards or away from the display to get different spatial frequencies and give a judgment. Results of the visual experiment are shown in Table 5 .

Table 5. Results of the visual experiment in which six students judged whether illuminance distributions in Fig. 8 and Fig. 9 were uniform

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Symbol “√” or “×” denotes uniform or nonuniform pattern was observed respectively. Three symbols in a cell denote observation on the initial position, being moved towards the display and away from the display orderly.

All the figures, especially those achieved through CSF method, were judged to be uniform by most of the observers with designated viewing condition, though few (observer 2) who may have individual CSF significantly departing from Barten model could perceive nonuniformity. Noticeably, nonuniformity was observed in figures corresponding to CSF method if the observer was moved from the initial observation position away from the display. This is due to that spatial frequencies of all the distributions in Fig. 8 and Fig. 9 are smaller than about 5cycle/degree where CS_th increases with the increasing of spatial frequency. If the observer is moved away, larger viewing distance that indicates larger spatial frequency will lead to greater sensitivity of human vision system and nonuniformity will be perceived. On the other hand, if the observer is moved closer, weaker sensitivity of human vision system will result in a uniform distribution observed all the time.

It can be revealed from Fig. 8 and Fig. 9 that parameters calculated through SC and CSF method can both produce uniform lighting. However, parameters from CSF method are more “critical”, in other words, under the same given parameters, optimum system thickness is smaller or optimum LED pitch is larger. So it is demonstrated that CSF method can achieve more appropriate design parameters in comparison with SC method in case that human vision effect is considered.

4. Application of CSF method in LALS with non-Lambertian LEDs

Critical uniformity condition problem of LALS-Lambertian has been successfully solved with the aid of CSF method. However, sometimes other types LEDs are utilized to construct an LALS for high luminance, larger LED pitch, thinner system and so on. It is necessary to investigate critical uniformity condition for LALS with non-Lambertian LEDs.

LUXEON^® Batwing LED from Lumileds Philips, as a conventional non-Lambertian LED, was selected for investigation for example. Typical analytical expression of its SRD is given in Eq. (8) and a polar distribution curve is shown in Fig. 10 [19].

 

Fig. 10 Polar SRD curve of LUXEON^® Batwing LED

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where P₁ = 0.386, P₂ = 0.08, P₃ = 0.369, P₄ = 0.55, P₅ = 0.671, P₆ = 0.153 and θ is in radius.

In an LALS with 10 by 10 foursquare arrangement and LUXEON^® Batwing LEDs, which is denoted as LALS-Batwing, LED pitch has been determined to be 10mm, and then similar study was implemented except I(θ,φ) was substituted by Eq. (8) but not that of Lambertian LED. To acquire optimum system thickness, differences between CS of $\overline{E}(x,y_k)$ that corresponds to each characteristic line and their thresholds under z from 1mm to 30mm are shown in Fig. 11 .

 

Fig. 11 Differences between CS of $\overline{E}(x,y_k)$ corresponding to each characteristic line and their thresholds under different system thicknesses z in LALS-Batwing with d = 10mm. z₁, z₃ and z₅ are thicknesses those can lead to the minimal average value of all the curves at troughs; z₂ is the thickness that can lead to the maximal average value of all the curves at a peak; z₄ and z₆ are thicknesses those can make all the curves just above the zero line.

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Something different from LALS-Lambertian were found in Fig. 11. The curves are remarkably oscillatory with several troughs and peaks in comparison with that in Fig. 5, which results in multiple critical system thicknesses and implies non-monotonic uniformity varying with system thickness. Critical system thicknesses z₄ and z₆, which are determined as the thickness that can just make all the curves above the zero line, along with other four thicknesses z₁-z₃ and z₅ corresponding to troughs and peaks, which are determined as the thickness that can lead to the minimal average value of all the differences at a trough (z₁, z₃ and z₅), or maximal average value of all the differences at a peak (z₂), are all marked with black circles in Fig. 11. To verify the variation behavior of uniformity with system thickness, visual illuminance distributions corresponding to z₁ to z₆ are shown in Fig. 12 , along with the average of differences between CS of six characteristic lines and respective thresholds, as denoted as ΔCS_avg. Additionally, coefficient of variation of root mean square error (CV (RMSE)), which has been proven a practical metric for image [6, 20], of the central grid is adopted to assess the uniformity supplementarily, as given in Eq. (9). All the CV (RMSE) corresponding to z₁ to z₆ are also provided in Fig. 12. If the observer is made 8.86cm away from the computer display as each sub figure horizontally covers 2.48cm, it can be seen that critical uniformity occurs twice at z₄ and z₆. Moreover, uniformity varies non-monotonically from z₁ to z₆, which is abnormal with regard to general design experience. Finally, z₄ beat the larger critical thickness z₆ and was determined to be the desired optimum system thickness for this problem. In fact, z₂ can be also suitable for some applications with lower uniformity requirement.

 

Fig. 12 Visual illuminance distributions of LALS-Batwing with d = 10mm corresponding to z₁ to z₆ in Fig. 13 respectively. ΔCS_avg and CV (RMSE) are also provided for each sub-figure.

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where $\overline{x}$is mean value and σ is standard error of 10 by 10 illuminance data points uniformly sampled in the central grid.

The abnormal non-monotonic uniformity has been also simply demonstrated through SC method in our former research [4, 5], in which multiple zero points of second derivative of illuminance distribution function on a point were found. In Fig. 13 , d = 10mm, second derivative of illuminance distribution function varying with z at point (0,0) is shown, simultaneously, differences between CS of $\overline{E}(x,y_0)$ and its threshold varying with z are also shown for comparison. The same comparison at point (0,d/2) is shown in Fig. 14 .

 

Fig. 13 Second derivative of illuminance distribution function varying with z at point (0,0) (upper sub figure) and differences between CS of $\overline{E}(x,y_0)$ and its thresholds varying with z (down sub figure)

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Fig. 14 Second derivative of illuminance distribution function varying with z at point (0,d/2) (upper sub figure) and differences between CS of $\overline{E}(x,y_5)$ and its thresholds varying with z (down sub figure)

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In Fig. 13 and Fig. 14, for every zero point achieved through SC method, a close zero point can be always found through CSF method, which congruously reveal change law of the illuminance uniformity. However, great advantages of CSF method can be demonstrated in comparison with SC method. Firstly, SC method can only judge whether it is uniform while CSF method can provide how uniform it is with the aid of the value of CS. More important, SC method can be only used at a point to determine “locally uniform” while CSF method can be used on a line. Specifically, in LALS-Lambertian, single zero point corresponds to each characteristic point and the maximal one can be empirically determined to be the optimum one. However, in LALS-Batwing multiple zero points at every characteristic point are found, wherein zero points with the same order and close value corresponding to different characteristic points are called “homogeneous zero points”. It is hard to determine whether the distribution become critically uniform or only an oscillation of non-monotonic uniformity varying with z occurs around a series of homogeneous zero points. Moreover, no experience can tell which one among a series of homogeneous zero points should be determined to be the optimum one even critical uniformity occurs. On the other hand, CS of all the characteristic lines can be considered at the same time with the aid of CSF method and change law of the uniformity can be completely demonstrated. For instance, z₂ calculated through SC method in Fig. 13 and Fig. 14 (in fact z₂ corresponding to more characteristic points can be calculated) imply that critical uniformity may occur around z = 10mm, but CS calculated through CSF method in Fig. 11 demonstrate that no value of z can make CS corresponding to all the characteristic lines larger that the threshold and the distribution will keep nonuniform..

Analogously, if z is determined to be 20mm, differences between CS of each $\overline{E}(x,y_k)$ and their thresholds under d from 1mm to 50mm is shown in Fig. 15 . And visual illuminance distributions of LALS-Batwing with z = 20mm corresponding to d₁ to d₆ as well as their ΔCS_avg and CV (RMSE) are shown in Fig. 16 respectively. It can be revealed that d₃ = 11.12mm, which is the largest LED pitch that can just make all the curves above the zero line, is the desired optimum LED pitch. Moreover, illuminance uniformity varies with LED pitch non-monotonically.

 

Fig. 15 Differences between CS of $\overline{E}(x,y_k)$ corresponding to each characteristic line and their thresholds under different LED pitches d in LALS-Batwing with z = 20mm. d₁ and d₃ are LED pitches those can make all the curves just above the zero line; d₂, d₄ and d₆ are LED pitches those can lead to the minimal average value of all the curves at a trough; d₅ is the LED pitch that can lead to the maximal average value of all the curves at a peak.

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Fig. 16 Visual illuminance distributions of LALS-Batwing with z = 20mm corresponding to d₁ to d₆ in Fig. 15 respectively. ΔCS_avg and CV (RMSE) are also provided for each sub-figure.

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In the analysis above, critical uniform condition for LALS constructed with Lambertian or non-Lambertian LEDs were studied through two representative systems, from which design method of optimum structural parameters can be summarized while designing an LALS with uniform lighting:

5. Future work

To achieve high quality uniform lighting produced by LALS, design of uniform brightness or illuminance distribution on the target plane is the first and the most important step. Next, uniform planar and spatial color distribution on the target plane should be considered [21, 22]. How to quantitatively calculate planar and spatial color distribution and then select an appropriate uniformity criterion with taking account of human vision effect will be carefully considered in our future work.

6. Conclusion

In this paper, contrast sensitivity function theory was adopted to achieve critical uniform lighting condition for LED array lighting system by regarding illuminance distribution in one dimensional direction as a sinusoidal/cosinoidal luminance modulated grating. Two representative systems constructed with Lambertian and Batwing type LEDs were studied. Resultant optimum structural parameters achieved through CSF method are demonstrated to be more appropriate than those achieved through SC method, which can lead to better practicability, lower cost, more attractive appearance and so on. Moreover, an abnormal phenomenon that uniformity of LALS with non-Lambertian LEDs varies with structural parameters non-monotonically was found and analyzed. A practical design method of LALS with uniform lighting was summarized by taking this abnormal phenomenon into consideration. In the last, reversing design from structural parameters to LED’s spatial radiation distribution was simply studied. Proposed method can be widely used in LALS design to save simulation or experiment time, or even solve reversing design problem that is hardly possible to be dealt with tentative simulations or experiments.

Appendix

In the appendix section, application of CSF method in perceiving square wave will be introduced and it can give a reference to the application of CSF method in judging uniform lighting produced by an LALS [10, 11, 16, 17].

A square wave S(x) with a cycle of T, amplitude of A and mean of X, as shown in Fig. 17 , can be decomposed into Fourier series, as given in Eq. (10). Amplitude of the 2n-1 order harmonic is 4A/[(2n-1)π] where n is positive integer. Accordingly, CS corresponding to the 2n-1 order harmonic is Xπ(2n-1)/(4A), as 1:3:5:7…. Obviously, CS corresponding to the fundamental harmonic is the smallest among them. And then, when CS corresponding to the fundamental harmonic just reaches CS_th corresponding to the fundamental spatial frequency (the same as the spatial frequency of S(x)), CS corresponding to other harmonics will be usually larger than CS_th corresponding to their spatial frequencies respectively i.e. CS corresponding to other harmonics will be upon the curve in “uniform region” in Fig. 18 , as marked with blue triangles in Fig. 18. Considering that human vision system is approximatively a linear system and stimulus of a square wave can be regarded as sum of stimuli of all harmonics [10, 11], perception of the square wave can be determined by its fundamental harmonic. Then CS of the square wave can be calculated through its fundamental harmonic, as Xπ/4A instead of original X/A, which will be compared with CS_th corresponding to the fundamental spatial frequency to determine whether the square wave can be distinguished from uniform lighting.

 

Fig. 17 Square wave S(x) with a cycle of T, amplitude of 2A and mean of X

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Fig. 18 Two situations of application of CSF in perceiving square wave with the aid of Fourier series: (1) when spatial frequency of the square wave is quite large, it is accurate to use CS of the fundamental harmonic instead of that of the original square curve for the perception, as marked with blue triangles; (2) when spatial frequency of the square wave is quite small, it is inaccurate to use CS of the fundamental harmonic instead of that of the original square curve for the perception, as marked with red circles

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However, if spatial frequency of the square wave is quite small, as smaller than about 5cycle/degree, CS_th will increase with the increase of spatial frequency, which can be found in left of the curve in Fig. 18. As a result, when CS corresponding to the fundamental harmonic reaches CS_th corresponding to the fundamental spatial frequency, those of higher order harmonics may be also around the curve, as marked with red circles in Fig. 18. Then it is inaccurate to use CS of the fundamental harmonic instead of that of the original curve for the perception of the square wave and a more complex calculation module should be adopted.

Both situations, as accurate and inaccurate to use CS of the fundamental harmonic instead of that of the original square curve for the perception of the square wave, have been verified by former researchers [16, 17].

(10)$$S(x)=X+{\displaystyle \sum _{n=1}^{\infty }\frac{4A}{(2n-1)\pi }\sin \frac{2\pi (2n-1)x}{T}}$$

Acknowledgments

This work was supported by NSFC Key Project under grant number 50835005, NSFC Project under grant number 50876038, High Tech Project of Ministry of Science and Technology under grant number 2009AA03A1A3 and Guangdong Real Faith Optoelectronics Inc.

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