## Abstract

This paper presents a methodology analogous to a general lens design rule to optimize step-by-step the spectral power distribution of a white-light LED cluster with the highest possible color rendering and efficiency in a defined range of color temperatures. By examining a platform composed of four single-color LEDs and a phosphor-converted cool-white (CW) LED, we successfully validate the proposed algorithm and suggest the optimal operation range (correlated color temperature = 2600–8500 K) accompanied by a high color quality scale (CQS > 80 points) as well as high luminous efficiency (97% of cluster’s theoretical maximum value).

© 2011 OSA

## 1. Introduction

Light-emitting diode (LED) technology has profoundly changed the way light is generated across a wide field of applications due to its unique characteristics, including possibly the highest optoelectronic conversion efficiency as well as the capability of modulating spectral composition and environmentally benign raw materials [1]. Among these features, one challenge in the design of a LED-based cluster is how to adjust the spectral power distribution (SPD) in an underdetermined condition, thus enabling us to manipulate strategically the chromaticity point, light quality, and system efficiency according to different operational purposes. For example, we are able to enhance the fidelity appearance in high-color-quality mode or to employ higher efficiency at a sacrifice of color rendering in an unoccupied area [2].

The mixing question for a white LED cluster can be separated into three aspects:

- (a) Energy––the most widespread figures of merit from the viewpoint of energy are the luminous efficacy of radiance (LER) and the luminous efficiency (LE). The LER represents the amount of luminous flux (lumen) converted from a per-unit optical power (watt), whereas the LE is defined as the luminous flux normalized to the electrical input power (watt) expended to operate the LED. In principle, the LE is the product of the LER and electric-to-optical power conversion efficiency [3]. In order to approach the relationship in terms of efficiency and color rendering, A. Žukauskas
*et al.*found an optimal boundary (Pareto front) to address the fundamental tradeoff between the LER and the color rendering index (CRI) via an LED-primary-based approach [4]. The optimal boundary subject for one artificial SPD has the potential to provide a useful guide in the design of a polychromatic system. To date, G. He*et al.*adopted a more practical index, LE, as a merit figure and transferred this concept into laboratory practice, where different LED white composite spectra were analyzed and realized over a range of color temperatures [5,6]. - (b) Light quality––the major characteristic of white light quality is its ability to reproduce colors of illuminated objects with high fidelity, i.e., as close as possible to those perceived under sunlight or blackbody radiators. The CRI proposed by the CIE (Commision Internationale de l’Éclairage) is the most widely recognized figure of merit. However, CRI has been criticized for its lack of fidelity in ranking sources, especially those with highly peaked spectra such as LEDs [7]. One of the major deficiencies is the penalization of sources that produce high-chromatic saturation, which is actually preferred for human vision. As a consequence, numerous refinements are being explored, such as the color quality scale (CQS) [8], gamut area index (GAI) [9], and color saturation index (CSI) [10].
- (c) Mixing scheme––the SPD of an LED cluster can be synthesized by using (i) additive mixing of two or more single-color LED chips (LED-primary-based approach), (ii) wavelength-conversion via using phosphors or other materials (LED-plus-phosphor-based approach), and (iii) a hybrid approach composed of (i) and (ii) [11].

The prior SPD optimizations were addressed mainly via multiple single-color LEDs and usually had been restricted to certain specific conditions, such as CRI, LER, and so forth [4]. Although several cases using a hybrid approach have been proposed for color temperature adaptable systems [5,6], to our knowledge, there is a lack of general SPD synthesizing rules for practical LED clusters, which can systematically and efficiently optimize SPD for certain user-defined lighting qualities. In this paper, we make an attempt to borrow design techniques from a conventional lens system and offer a solution with wider operation windows to cover aforementioned environments. Our ultimate goal is to develop a general LED design procedure in a more complete treatment. The design flow in all respects can be closely analogous to a conventional lens design process that has long been developed by which the SPD of an LED cluster can be optimized by going through every step of the modeling. All the figures of merit affected by different factors are discussed, along with the experimental validation of an LED cluster that will be examined.

First, we emulate a single-color LED as a singlet, whose light-bending power determined by its curvature and refractive index can be conceptually analogous to the emitting luminous flux of an LED determined by the driving current and LE, respectively. As we mix a number of LEDs, the additive mixing by two single-color LEDs is equivalent to two singlet lenses. Likewise, the LED-plus-phosphor-based approach can be regarded as a cemented doublet (dichromatic) or triplet (trichromatic), depending on the number of emitting peak wavelengths. The concept is schematized in Fig. 1 . Based on the hypothesis outlined above, the SPD synthesis can be transformed into a classic lens design problem. For example, an LED cluster composed of red/cool-white/cool-white/green (R/CW/CW/G) is logically equivalent to a double Gauss lens system. The fundamental constraint such as diffraction limitation of a lens system is viewed accordingly as the theoretical boundary of the LER or CRI.

## 2. Concept of Design Procedure

The solution of a lens design is a typical inverse problem. Given the effective focal length (EFL) and degree of correction for an optical system, it is always possible to determine the curvatures, thicknesses, and number of lenses in sequence. For example, if we aim to design a lens system with a specified EFL and correct three Seidel aberration coefficients, it can be resolved analytically by a set with two singlet lenses; that leaves four degrees of freedom––two powers and two shape factors [the shape factor is defined as (R_{2} + R_{1})/(R_{2} − R_{1}), where R_{1} and R_{2} are the radii of the first and second surfaces, respectively]. Since the complexity of multiple lenses would increase the computational cost, a more efficient method in lens design would resort to an iterative process, as shown in Fig. 2(a)
.

Similarly, we adopt this idea by replacing the lens set with a number of LEDs for certain predefined environments, as proposed in Fig. 2(b). The design procedure includes six steps: (2.1) initial system, (2.2) define boundary condition, (2.3) optimization, (2.4) aberration or merit analysis, (2.5) judgment, and (2.6) tolerance analysis, and each step is discussed below.

#### 2.1. Initial System

Like a glass map in lens design, LED manufacturers offer a broad range of LED datasheets with available materials and peak wavelengths [12,13]. The dependence of LE on peak wavelengths can be analogous conceptually to a refractive index versus an Abbe number. It is known that the lens with the higher refraction index possesses higher bending power. Therefore, green- (505 nm) and amber- (595 nm) color LEDs would serve as appropriate candidates in the consideration of high LE, as shown in Fig. 3 .

If we plan to mix two single-color LEDs for a specific correlated color temperature, *T _{CC}*, the most straightforward solution is to select two complementary peak wavelengths on the chromaticity diagram. However, the question becomes more complex when multiple figures of merit are considered by a number of LEDs. To pick an appropriate LED set in a systematic way, we list three suggestions for the initial system [14]:

- 1. A mental guess. This way is workable for an expert, while it is laborious for a beginner.
- 2. A designed case from previous literature. It is the most common way to choose a design close to your requirements.
- 3. A search through the patent files. This is also time-consuming work, and consideration of avoiding the patent’s claims in your design is necessary.

At this moment, the second approach is easier to follow and, fortunately, many previous authors have disclosed their experience for specific qualities; e.g., the trichromatic source composed of primary emissions (630 nm, 530 nm, and 450 nm) makes surface colors appear more saturated, whereas the continuous spectrum that is designed to mimic daylight has better color-rendering ability [15]. In additional to the spectral aspect, the rapid progress in efficiency of phosphor technology and LED chip would drive the solid state lighting design toward more flexibility and combination [16].

#### 2.2. Define Boundary Condition

Before optimizing a predetermined initial system, the designer must define the domains of input variables. Such a step not only ensures a reasonable result but also reduces the computational time. In lens design, we usually set the curvatures and the thicknesses of lenses as the variables to be optimized, whose domains are mainly constrained by manufacturing feasibility. For an LED cluster, the variables in SPD synthesis would be the driving current (*I _{i}*) and the full width at half-maximum (

*Δλ*) for each LED component. Previous works have pointed out that variations of

_{i}*Δλ*could be characterized in terms of

_{i}*I*with consideration of the thermal effect in the LED chip [17]; therefore, in this paper, the driving current (

_{i}*I*) is chosen as the independent variable for setting the boundary condition.

_{i}The color mixing for such a condition can be described as in Eq. (1),

**t**represents the resulting tristimulus of an unknown input LED vector

**i**(

*M*-element) projected on the reference matrix

**T**[18]. If the system is critically determined (

*M*= 3), there exists a single solution (

*I*,

_{1}*I*,

_{2}*I*) to enable the resulting tristimulus (

_{3}*X*,

*Y*,

*Z*) as possibly closest to the target tristimulus (

*X*). Because the light response in human vision is relatively insensitive at a high luminance level, we can simply confine chromatic deviation Δxy < 0.01, which commonly is used in the lighting industry:

_{T}, Y_{T}, Z_{T}Once the system is underdetermined (*M* > 3), *I _{i}* has multiple results to satisfy Eq. (2). For such a case, in addition to the chromaticity point the physical limitation for each LED current

*I*should be imposed as in Eq. (3),

_{i}So far, **T** is assumed to be a constant reference matrix. However, the entries of the matrix would be a function of input current *I _{i}* .To include the dependence of

*I*, we then rewrite the column entry (

_{i}*X*,

_{i}*Y*,

_{i}*Z*) in Eq. (4) as

_{i}Reference [5] shows the dependence of *SPD _{i}*(

*λ*,

*λ*

_{0}_{,}

*,*

_{i}*Δλ*) on input current

_{i}*I*under a constant ambient temperature (

_{i}*T*). In order to fully characterize

_{a}*SPD*(

_{i}*λ*,

*λ*

_{0}_{,}

*,*

_{i}*Δλ*), we generalize the model that includes the influences of ambient temperature on

_{i}*λ*

_{0}_{,}

*and*

_{i}*Δλ*(Fig. 4 ). Accordingly,

_{i}*λ*

_{0}_{,}

*and*

_{i}*Δλ*, respectively, are no longer constants but functions of (

_{i}*I*,

_{i}*T*).

_{a}Both *λ _{0}*

_{,}

*(*

_{i}*I*,

_{i}*T*) and

_{a}*Δλ*(

_{i}*I*,

_{i}*T*) are a typical “two-input single-output” system, also called a surface fitting (SF) model [19]. The typical SF model can be decomposed by the basic functions

_{a}*f*(

_{j}**),**

*μ***is a vector with two inputs, e.g.,**

*μ**I*and

_{i}*T*, and

_{a}*ν*is the corresponding output, e.g.,

*λ*

_{0}_{,}

*or*

_{i}*Δλ*. The term

_{i}*a*is the

_{j}*j*th unknown coefficient. With appropriate basic functions, a set of sample data (

*μ**,*

_{k}*ν*)

_{k}

_{k}_{= 1···}

*by experimental results can be imported into Eq. (5) and expressed in a matrix form:*

_{m}Equation (6) is generally over-determined, *m* >> *n*, so that the coefficient vector **a** should be solved by minimizing the error function *E*(**a**) defined in Eq. (7) as

The *SPD _{i}*(

*λ*,

*λ*

_{0}_{,}

*,*

_{i}*Δλ*) is then obtained and usually is expressed by the Gaussian function. In addition, compensation terms,

_{i}*λ*

_{0}_{,}

*(*

_{i}*t*),

_{i}*Δλ*(

_{i}*t*), and

_{i}*SPD*(

_{i}*t*), could be attached to corresponding factors where device-aging dependence could be included.

_{i}#### 2.3. Optimization

For the lens design, it is likely to have identical EFLs due to a combination of different curvatures and thicknesses of the prescribed elements. Therefore, an additional mechanism of assessment, usually adopting a merit function, is necessary for the optimization process. Generally, the merit function of a lens system shall include the aberrations and should evaluate the impact of each parameter change on image quality. Similarly, different SPDs subject to different combinations of driving currents would result in the metamerism. A user-defined merit function is essential in order to consider the dependence of SPD on predefined performance. Equation (8) is an example of where we set the LE and CQS as the figures of merit. In principle, the merit function can be chosen arbitrarily and applied to multiple dimensions without loss of generality. Here we set the CQS as the merit for light-rendering capability. This is because it employs a set of color samples all of higher chroma and adopts a more uniform CIELAB color space than the CRI. The major improvements are that the CQS takes into account observer preferences by reflecting the differences between hue and saturation shifts and by using the rms of color differences to ensure that large shifts in any color sample can be adequately incorporated in the overall score.

Within the boundaries defined in Eqs. (2) and (3), as we change one of the driving currents, the merit function leads to a corresponding value. First, we randomly generate an initial population, *I _{pop}*, of “combination changes” by various driving currents that are similar to parameter changes in the lens system; a table showing merit function changes versus

*I*is obtained. After that, we bring the table into a global search engine, a continuous genetic algorithm (CGA), to achieve an improved spectral synthesis [20]. In the CGA, each row of the table is composed of a value of merit function, and the associated drive current combination is regarded as a chromosome. The chromosomes are ranked under a process of natural selection. The survivors are deemed fit enough to mate and to afford new offspring. By multiplying a certain mutation rate into the total number of drive currents, we are able to avoid the predicament of overly fast convergence or of being trapped by local limitation. Finally, the described process is iterated for several generations until the convergence requirement is satisfied. The precision of optimization is limited by the round-off error in the calculation.

_{pop}#### 2.4. Merit Analysis

The weight factor *w* provides an additional freedom for the user to determine the operation point among different figures of merit. Equation (8) constitutes a two-dimensional optimal boundary (Pareto front, PF) between the CQS and LE. Different weight values *w* profile a series of locus of operating points with different prescribed (*T _{CC}*,

*T*), as shown in Fig. 5(a) . For the sake of computation efficiency, we proposed two sampling methods, SA

_{a}_{1}and SA

_{2}, to analyze respectively the cluster performance among the CQS and LE. The concept of merit analysis is similar to the aberration analysis for different fields of view (usually at object height of 0, 0.5, 0.7, and 1) in lens design [21].

The principle of SA_{1} is based on the curve-fitting (CF) approach. The first step is to examine the locations of both extreme points (*P _{0}*,

*P*) at the optimal boundary, where

_{1}*P*(

_{0}*w*= 0) represents the efficiency mode where all the weight is attributed toward the LE.

*P*(

_{1}*w*= 1) thus represents the quality mode associated with all weighting CQS. If either end

*P*or

_{0}*P*is located at quadrant III, that means the CQS and LE fail to satisfy simultaneously the user-defined specification [quadrant I, defined by the minimum LE

_{1}_{m}and minimum CQS

_{m}]. The reason is due to the optimal boundary that always exhibits a tradeoff relation and is not likely to appear as a positive slope. If the Pareto front locus is profiled as PF

_{1}or PF

_{2}, we must interpolate other points such as

*P*and

_{a}*P*to help us succeed in fitting the optimal boundary curve. The CF process is the same as that of the SF model in Eqs. (5)–(7) but with a “single-input single-output” system. Taking PF

_{b}_{2}as an example, presently the modeled curve overlaps partially with quadrant I, defining an operation portion (red curve) with two extreme ends whose weights can be estimated by establishing another SF model,

*f*(

*P*)

*= w*, with four input points

*P*

_{0},

*P*,

_{a}*P*, and

_{b}*P*

_{1}as well as four output weights 0,

*a*,

*b*, and 1. As a consequence, an appropriate weight can be obtained from the proposed sampling method with a small number of sample points. The SA

_{1}procedure is summarized as shown in Fig. 5(b).

Compared with curve-fitting method SA_{1}, linear approximation is computationally efficient to determine the appropriate operating point, as shown in the dashed lines of Fig. 5(a). In this way, we assume $\overline{{P}_{0}{P}_{1}}$ already crosses quadrant I. For no particular reason, we choose *P _{0}* (the highest LE mode or efficiency mode) as the starting point. The increment rate of the CQS (

*CQS*

_{1/0}) at the expense of the LE decrement (

*LE*

_{1/0}) can be defined as in Eq. (9),

For an arbitrary point *P _{c}* (

*CQS*,

_{c}*LE*) located on the line $\overline{{P}_{0}{P}_{1}}$, the weight

_{c}*c*is determined by Eq. (10),

*c*also indicates the increasing rate of

*CQS*

_{1/0}as well as the decreasing rate of

*LE*

_{1/0}. Linear approximation SA

_{2}is a fast way to find the optimal operating point at the expense of a precise estimation of the weight value. Meanwhile, this method might face risk as in the PF

_{2}case that the real Pareto front curve is cross quadrant I, but the linear approximation $\overline{{P}_{0}{P}_{1}}$ case does not.

#### 2.5. Judgment

Up to this point, an optimal operation has been determined under an appropriate weighting value *w*. However, another aspect that must be taken into account in the framework of prototype is the margin analysis. During the course of spectral synthesis, in a situation where one LED is dimming to an extremely low level, we can possibly remove it without affecting system performance. Likewise, we can add the number of LEDs to allow the operation within adequate margins. Either scheme has good correspondence with the skill the lens design uses as follows [22]:

- 1. Among the operating wavelengths, add a new available wavelength and vary its emission power to analyze the merits (CQS and LE) again. It is usual to insert a wavelength at the large interval between peak wavelengths.
- 2. Replace two or more single-color LEDs by a phosphor-converted LED, or vice versa. If there is a remarkable performance advance in any kind of LEDs, try to adopt it.
- 3. Split an operating wavelength of too-high emission power into two adjacent wavelengths. This may be useful to avoid dangerous operation in a tiny margin of the requirements.

Once the optimal boundary crosses the user-defined specification (quadrant I), we can fix an appropriate weight value by using SA_{1} or SA_{2} for the merit function accordingly.

#### 2.6. Tolerance Analysis

Finally, the designer can introduce a small perturbation to each parameter sequentially (λ_{0,i}, Δλ* _{i}*,

*I*,

_{i}*T*) and observe the corresponding change. It is noted that the presented technique merely confines the discussion to the spectral range––it is not likely to predict the light field changed by the geometric deviation such as an LED package error or assembly misalignment. A possible compensation mechanism that constantly measures the SPD on the illuminated plane and gives feedbacks to drive currents might be helpful to improve the tolerance margin in the LED cluster [3].

_{a}## 3. Design Example

In order to validate the devised model, here we setup a platform of a pentachromatic white source composed of four single-color LEDs and a phosphor-converted CW LED (Excellence Opto. Inc., EOQ5P), respectively. With ambient temperature *T _{a}* = 300 K and driving currents

*I*= 20 mA, the LED spectra are shown in Fig. 6 . An adequate layout of an LED arrangement and optics by a first-order design was considered to deliver a uniform illumination [23]. Because of a low level of driving currents, the SPD modeling for each color LED can be assumed to satisfy the scalability and addictivity in a color mixing scheme [24].

To fulfill the modeling through aforementioned Sections 2.2–2.6, four operational SPDs with two extreme points (*P _{0}*,

*P*) under

_{1}*T*= 3000 K and 6500 K are verified, as shown in Fig. 7 . The simulation results are in close agreement with the experimental measurements within Δxy = 0.01 chromaticity deviation. Figure 8 features illuminant environments for different color temperatures (

_{CC}*T*= 3000 K and 6500 K), where the composite spectra from the LED matrix are digitally controlled by pulse-width modulation (PWM) with 128 gray levels.

_{CC}In addition to experimental validation, more insight can be pursued for smart lighting operation. Here we assume the minimum requirements for color rendering CQS_{m} = 80 points and LE_{m} = 60 lm/W. Based on the linear approximation SA_{2} method, the loci of R/G/B, R/G/B/A, and R/G/B/A/CW are plotted in Fig. 9
. The black curve in Fig. 9(a) depicts the referenced single solution of an R/G/B cluster for each color temperature. By adding amber (A) to the R/G/B cluster, we can improve an average of 50% CQS without too much loss of LE. The result is generally in agreement with the concept that a wide spectrum would improve the color-rendering performance. On the other hand, the contribution of an additional CW LED is depicted in Fig. 9(b), which could further increase 5% in CQS and 20% in LE over the full range of color temperature. The reason is due to the fact that a CW LED associated with high efficiency offers a good option to replace the function of blue color. The detail will be analyzed at the end of this section.

The following is to determine both operating windows for R/G/B/A and R/G/B/A/CW clusters. We first set point *P _{0}* (w = 0) that lies in the efficient mode as the starting point for each

*T*. Figure 10 shows the information about

_{CC}*CQS*and

_{1/0}*LE*that indicates the increment rate of CQS at a sacrifice of the decrement rate of LE in

_{1/0}*P*. Because the R/G/B/A cases for all

_{0}*T*are located at the right top corner, the designer would undoubtedly chose a high weight value (

_{CC}*w*~1) to boost the color rendering ability at little expense in cluster efficiency. This action is equivalent to drive

*P*approaching

_{0}*P*along the straight line in Fig. 9(a). Nevertheless, the R/G/B/A cluster still suffers a stringent operating window of

_{1}*T*(2800–3000 K), which precludes its use in intelligent lighting applications.

_{CC}Compared with the R/G/B/A cluster, the addition of a cold-white (CW) LED extends the operation window throughout the entire color temperature range. To prove this we select starting point *P _{0}* at

*T*= 3000 K, where

_{CC}*CQS*= 66.8 points (unqualified) and

_{0}*LE*= 66.7 lm/W [refer to Fig. 9(b)]. The correspondin information for

_{0}*CQS*= 34.3% and

_{1/0}*LE*= −2.4% at the same point

_{1/0}*P*can be found in Fig. 10. It is easy to take the above parameters into Eq. (10) and derive an appropriate weight of 0.79 to fulfill the requirement via increasing the CQS value to 85 points at a sacrifice of 1.3 lm/W. Generally, the weighting value can be conducted to compare between

_{0}*CQS*and

_{1/0}*LE*. That means the balance condition of

_{1/0}*CQS*

_{1/0}≈−

*LE*

_{1/0}at

*T*of 5200 K in Fig. 10 can be regarded as a turning point for the weight selection. Taking the R/G/B/A/CW combination for example, it is logical to approach the requirements by setting a large weighting value (

_{CC}*w >*0.5) for

*T*< 5200K, and vice versa.

_{CC}At this point we can successfully determine the operation point with the proposed methodology and set an optimal lighting environment for an R/G/B/A/CW system as shown in Fig. 11
. The operation window is extended to span 2600–8500 K with user-defined requirements of CQS_{m} = 80 points and LE_{m} = 60 lm/W, which would shrink under more severe lighting requirements accordingly (e.g., the operation window of 3200–5600 K for CQS_{m} = 90 points and LE_{m} = 64 lm/W).

Based on Fig. 11, we find that when the correlated color temperature is less than 6400 K, the power ratio is mainly governed by the light quality requirement (high weighting factor), and each component has a comparable amount. On the other hand, when the operation temperature is higher than 6400 K, the efficiency requirement (high weighting factor) is dominated and contributed to by a CW LED. The combination of LED clusters reduces to R/G/CW for 6400 K < *T _{CC}* < 8500 K as shown in Fig. 11(b). Within the operation window of 2600–8500 K, the result in Fig. 11(b) also indicates the function of the blue LED has been replaced by the CW light, so we can discard it from the cluster for most general lighting applications.

## 4. Conclusion

A novel LED mixing scheme analogous to the conventional lens design process has been proposed. The algorithm enables the users to determine easily the optimal LED setup to meet requirements such as light efficiency, color quality, or other figures of merit over a wide range of color temperatures. The procedure includes six steps––(2.1) initial system, (2.2) define boundary condition, (2.3) optimization, (2.4) merit analysis, (2.5) judgment, and (2.6) tolerance analysis, and each step has been considered and validated in detail by an experimental platform. The design example of an R/G/A/CW cluster can extend the operation window to 2600 K < *T _{CC}* < 8500 K for the requirements of CQS

_{m}= 80 points and LE

_{m}= 60 lm/W. Due to its simplicity and versatility, the proposed technique certainly has a promising impact on rapid prototyping and other specialized features for lighting applications.

## Acknowledgment

The authors thank Mr. SB Chiang and SM Tasi for their technical support and discussion. This work was financially supported by the National Science Council, Taiwan, under grant NSC 99-2221-E-009-067-MY3.

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