The spectral power distributions of tri- and tetrachromatic clusters of Light-Emitting-Diodes, composed of simulated and commercially available LEDs, were optimized with a genetic algorithm to maximize the luminous efficacy of radiation and the colour quality as assessed by the memory colour quality metric developed by the authors. The trade-off of the colour quality as assessed by the memory colour metric and the luminous efficacy of radiation was investigated by calculating the Pareto optimal front using the NSGA-II genetic algorithm. Optimal peak wavelengths and spectral widths of the LEDs were derived, and over half of them were found to be close to Thornton’s prime colours. The Pareto optimal fronts of real LED clusters were always found to be smaller than those of the simulated clusters. The effect of binning on designing a real LED cluster was investigated and was found to be quite large. Finally, a real LED cluster of commercially available AlGaInP, InGaN and phosphor white LEDs was optimized to obtain a higher score on memory colour quality scale than its corresponding CIE reference illuminant.
© 2011 OSA
Over the past few years, interest in solid-state lighting as an alternative to more conventional light sources has grown dramatically due to benefits such as the increased (and still increasing) luminous efficacy and hence the potential of substantial energy savings and environmental benefits, the compactness of the light source, the lifetime, the tunability of the spectral power distribution to create and/or enhance the mood of a room, … [1–4].
Another aspect of lighting, important to many end-users, such as lighting designers, architects, shop and retail owners, …, is colour quality. Currently colour quality is evaluated by the CIE colour rendering index as the ability of a light source to render the colours of a set of illuminated test colours as closely as possible to their colours under a blackbody radiator or a daylight illuminant. Unfortunately, this method does not always correspond well with the actual perceived colour quality of a white light source [5–9]. Reasons might be several, but in the authors’ opinion, one of the main reasons is that, for many users, colour quality is more than just fidelity with a CIE reference illuminant. Other aspects such as preference, attractiveness, colour discrimination, colour harmony need to be taken into account [10, 11]. Over the past decades, several metrics dealing with these additional aspects have been presented: e.g. Judd’s flattery index , Thornton’s preference index , colour quality scale , colour saturation index , gamut area index , colour harmony index , feeling of contrast , memory colour quality metric [19, 20], …
Colour quality metrics, like those above, and the CIE colour rendering index, are usually used to assess the colour quality of an existing light source. However, applied in reverse, they can also be used to calculate spectral power distributions with good colour quality as assessed by that particular colour quality metric. Lamp manufacturers have been doing this ever since the CIE colour rendering index became a standard metric. Unfortunately for many lamp manufacturers, it transpires that the CIE colour rendering index does not always correspond well with visual appreciation, especially for solid-state light sources [5–9, 20]. Many users are often also more interested in how good objects look under a given light source than they are in the colour fidelity with a CIE reference illuminant. Saturation or chroma enhancement is often reported as having a positive influence on perceived colour quality . Zukauskas et al.  used their colour saturation index to optimize trichromatic LEDs for increased saturation ability. However, no restrictions were put on their hue distortion index, which was found to be highly correlated with the colour saturation index, resulting in a possibly unacceptable colour appearance under the optimized LED cluster. Indeed, large chroma shifts are accompanied by large shifts in hue, as was reported by Davis et al. .
A metric that does not depend on chroma/saturation enhancement, but that does depend directly on visual appearance ratings of a set of familiar objects, is the memory colour quality metric developed by the authors [19, 20]. The basic idea of the metric is simple: the better the colour appearance of an object under a light source resembles what is expected, the better will be the perceived colour quality of the light source. This metric has also been shown to correspond well with visual quality assessments from three different studies, all of which included several solid-state sources [20, 22, 23]. In this work, the spectral power distributions of LED clusters will be optimized with respect to the memory colour quality metric.
2.1. Memory colour quality metric
In a previous study, the authors investigated the colour appearance ratings of a set of real familiar objects with colours distributed around the hue circle . The objects were presented in approximately one hundred different colours to a group of observers by illuminating them inside a specially constructed LED illumination box. Any clues to the colour of the illumination were masked, creating the illusion that the objects themselves changed colour. The observers were asked to rate the similarity of the perceived object colour to their idea of what the object should look like. The pooled colour appearance ratings were modelled in the uniform IPT colour space  by a modified bivariate Gaussian distribution.
Based on these similarity distributions, the colour quality of a light source is estimated as follows . First, for all objects, the tristimulus values under the light source are calculated using the spectral reflectance of the objects and the CIE 10° standard observer. Secondly, the corresponding tristmulus values are calculated under D65, the IPT white point, with the CAT02 chromatic adaptation transform. Thirdly, the corresponding tristimulus values are transformed to IPT chromaticity coordinates, X i = (P i,T i). Fourthly, the function values of the corresponding similarity distributions S i(X i) are calculated with the object chromaticities X i as input, resulting in a set of ten S i values describing the degree of similarity with each object’s memory colour:
The model parameters a i,1-5 describe the location, shape, size and orientation of the similarity distribution S i(X i). Finally, the general degree of memory colour similarity S a, assumed to be a measure for colour quality, is obtained by taking the geometric mean of the ten individual S i values.
2.2. Parametric LED models
The spectral power distribution of a single colour or a phosphor type LED can be modelled based on physical principles . However, such models require many parameters making them less suitable for optimization, especially in the case of clusters composed of several LEDs. The spectral power distribution of a single-colour LED is therefore often approximated by a simple Gaussian function. However, to correct for the large shoulder of a Gaussian function Ohno et al.  proposed a more accurate representation Φe, λ,c:
The spectral power distribution of a phosphor-type white LEDs is typically composed of a peak at a short wavelength and a broad hump at medium-to-long wavelengths. The hump is produced by the conversion of a portion of the light emitted by the short wavelength LED (the pump) by one or more phosphors. Phosphors can basically be modelled by a Gaussian. Based on the model for a single-colour LED proposed by Ohno et al. , the following set of equations was used to simulate the spectral power distribution Φe,λ, ph of phosphor-type LEDs with two phosphors:
With λ0,c and ∆λ0.5,c the peak wavelength and the spectral width of the short wavelength LED; λ0,ph,i and ∆λ0,phi (i=1,2) the peak wavelength and spectral width of the two phosphors. The factor α, controls the height of the phosphor hump (relative to the pump) and varies between approximately 0.5 and 2 for common phosphor white LEDs . The factor β, the relative contribution of the two phosphors, varies between 0 and 0.5. The multiplication at the end with the function is to ensure that the spectral power distribution of the phosphor white LED has a sharp shoulder at the short wavelength side. For a monochromatic LED, 3n parameters are needed; for a phosphor white LED, 9m parameters. The spectral power distribution of a LED cluster composed of n coloured LEDs (i=1..n) and m phosphor-type LEDs (j=1..m) can therefore be represented in a (3n+9m)-dimensional parametric space Ψ as a vector:
With φ the fluxes of the individual LEDs in the cluster.
In general, optimization involves searching the optimization domain to find the global solution X o that maximizes one or more, possibly conflicting, objective functions f i (X). In this work, the optimization domain was the (3n+9m)-dimensional parametric space Ψ, where each point X represents a LED cluster described by Eq. (5). The optimization problem was characterized by several objectives. The first one being optimal colour quality. In this work, the basic assumption was that a pleasing colour quality closely corresponds to a high degree of general similarity of a set of familiar objects with their memory colours, i.e. a high value of S a. As a second objective, the luminous efficacy of radiation or LER was also taken into account to ensure that the optimization did not result in a very inefficient spectral power distribution Φ e,λ. The LER is defined as:27]. A Pareto optimal solution is optimal in the sense that improving one objective would degrade the performance for at least one other objective. This means that without any further information, one of these Pareto optimal solutions cannot be regarded as better than any other one.
2.3.1 Genetic algorithms
Although there are several global optimization algorithms available in the literature, in this work, genetic algorithms (GA)  were chosen because they are able to scan a vast set of solutions, they do not depend on a starting solution, they are very useful for complex problems, and most importantly, they can be easily modified to estimate the Pareto optimal set. A GA is based on the Darwinian principle of survival of the fittest. The algorithm encodes the decision variables X in a set of genes, called the chromosome or genome. It then stochastically generates a population of individuals, i.e. candidate solutions, each with its own genotype. The population is then allowed to evolve into new generations. The parents for each new generation are stochastically selected based on the “fitness” of the individuals as measured by the objective function. After enough generations an estimate of a global solution is obtained. For multi-objective problems, where there might not be one optimal solution, the single-objective GA is modified to evolve towards the Pareto optimal front. Several multi-objective evolutionary algorithms (MOEA) are described in literature and a good overview can be found in Coello and Lamont  and Konak, Coit Smith . The MOEA selected in this work is the non-dominated elitist NSGA-II genetic algorithm, a widely accepted benchmark in the MOEA research community . A complete description of the NSGA-II algorithm is beyond the scope of this article, and the interested reader is kindly referred to the paper by Deb, Pratap, Agarwal and Meyarivan .
3.1. LED models
In this work, the optimization involved tri-and tetrachromatic LED clusters. Trichromatic clusters were composed of a Red LED, a Blue LED, and a Green or a phosphor type LED; tetrachromatic clusters were composed of a Red LED, a Green LED, a Blue LED, and a Yellow-to-Amber or a phosphor type LED. The decision variable space was therefore reduced to 9 and 12 dimensions for clusters composed exclusively of single coloured LEDs (resp. R/G/B and R/G/B/Y-A); and to 12 and 18 dimensions with the incorporation of one phosphor type LED (resp. R/B/phLED and R/G/B/phLED). An estimate of the Pareto optimal set for the multi-objective problem described in section 2.3 was obtained using the NSGA-II algorithm. The GA ran for 250 generations with a population of 10000 individuals. The S a -LER Pareto optimal fronts for the 4 types of LED clusters are shown in Fig. 1 .
From Fig. 1 it is clear that as far as the level of achievable colour quality is concerned, a replacement of the green and yellow-to-amber LEDs with a phosphor type LED in the tri- and tetrachromatic clusters did not have a substantial impact. The highest colour quality achievable (as measured by the Sa-metric) was approximately 0.845 for both the tri- and the tetrachromatic clusters. In the case of the trichromatic clusters the replacement did have a small effect on the luminous efficacy of radiation: slightly lower LER values were observed for the same colour quality or S a values. This slight decrease is most likely due to the broader spectral width of the phosphor type LED in comparison with the green LED. For the tetrachromatic clusters, no difference in LER was found for S a ≥ 0.678. Tetrachromatic clusters had a higher LER than trichromatic clusters, although this difference was reduced to zero at higher colour quality (S a ≥ 0.755, i.e. the maximum Sa-value obtainable for CIE illuminants); except for the clusters containing a phosphor type LED where a small difference was still observed at the highest achievable Sa values. Tetrachromatic clusters of the R/G/B/Y-A type had the highest possible LER (534 lm/W), but at significantly reduced colour quality (S a = 0.473). Overall, the differences in performance of the four simulated types of LED clusters were found to be small to non-existent. The performances of the LED clusters was also compared to those of a set of standard CIE reference illuminants (Planckian radiators: 2700 K, 3000 K, 3500 K, 4000 K, 4500 K and daylight phases: 5000 K, 5500 K, 6000 K, 6500 K, 7000 K, 8000 K). As is clear from Fig. 1, the optimized LED clusters performed substantially better than the CIE reference illuminants in terms of both luminous efficacy of radiation and colour quality as measured by the S a metric, emphasizing the potential of LEDs for creating energy-efficient light sources with good colour quality.
Optimum peak wavelengths and spectral widths of the different LEDs in each cluster were determined by investigating the subset of Pareto optimal solutions with a colour quality above that of the highest of the CIE reference illuminants (S a ≥ 0.775). A histogram of the peak wavelengths of the LEDs, shown in Fig. 2 , revealed that the blue and red LEDs had approximately the same optimal wavelengths of respecively 460 nm and 612 nm for all types of LED clusters. In the case of the trichromatic LEDs, the peak wavelength of the green LED or the phosphor hump was found to be 539 nm. The same optimum peak wavelength was found for the green LEDs in the tetrachromatic LEDs. The yellow-to-amber LED showed three distinct optimal peak wavelengths; one at 560 nm; one at 594 nm and another around 604 nm, close to the optimal peak wavelengths of the red LEDs. The phosphor type LEDs of the tetrachromatic cluster also had three distinct optimal peak wavelengths. The first peak (534 nm) (approximately) coincided with the peak of the green LEDs. The second peak was located close to the optimal peak wavelength of the yellow-to-amber LED at 552 nm. A third peak was found at 604 nm, near the optimal peak wavelengths of the red LEDs. The short wavelength pumps of the phosphor type had their optimum peak at approximately that of the blue LEDs, but they also showed a separate optimum peak at approximately 485 nm. The fact that peaks with a relatively narrow spread showed up in the histograms and especially that the same wavelengths recurred in the different types of clusters suggests that these wavelengths are of special importance in obtaining a spectral power distribution with a good colour quality and luminous efficacy. The 552 nm and 560 nm optimum wavelengths of the phosphor type LED and the yellow-to-amber LED in the tetrachromatic clusters can be easily explained as peak wavelengths that maximize the luminous efficacy of radiation, because they are located close to the peak wavelength of the CIE 1924 luminous efficiency function. No such peaks were however found for the trichromatic clusters. The closest peak was found at 539 nm, which is very close to one of Thornton’s prime colours. Prime colours are those spectral lights for which the human visual system is most sensitive . In fact, over half of the optimum peak wavelengths (460 nm, 539 nm, 604 nm and 612 nm) found were located close to one of Thornton’s three prime colours, i.e. 450 nm, 533 nm and 611 nm . This is interesting, because, although they are associated with optimum human visual sensitivity, they are not used in the calculation of the luminous efficacy. This suggests that they are probably linked to colour quality. Finally, two more optimum wavelength peaks (485 nm and 594 nm) were found to be important, but solely for colour quality, as they were not associated with the peak of the luminous efficiency function. Their location suggests that they served to fill in the gaps in the cyan and the orange region of the spectral power distributions.
The same analysis was performed on the Full-Width-Half-Maxima of the different LEDs. The optimal FWHMs of all LED types for each cluster are given in Table 1 .
From Table 1, it is immediately clear that the optimum widths of the red, green and blue LEDs are approximately the same for each type of cluster. The spectral width of the phosphor type LEDs did show a rather large difference. The lack of a narrow peak at medium wavelengths (green LED) decreased the width of the phosphor type LED in the trichromatic cluster, suggesting that a rather narrow peak at medium wavelengths is required to obtain an optimal colour quality. In the case of trichromatic clusters, a replacement of the phosphor type LED by a green LED, with an even narrower spectral width, also resulted in a small increase in colour quality (at the same luminous efficacy).
3.2. Commercially available LEDs
In this section, the feasibility of designing a practical tri- or tetrachromatic LED cluster is investigated.
Previously, it was found that all cluster types, except for the R/B/phLED type, could achieve the same high colour quality. Tetrachromatic clusters had a slightly higher luminous efficacy, but this advantage disappeared at higher colour quality. Therefore, a LED cluster with good colour quality could, in principle, be constructed from any of the previously trichromatic simulated cluster types. Unfortunately, the optimization of a LED cluster composed of commercially available LEDs is restricted mainly for two reasons. First, mass produced LEDs are not available at every peak wavelength and every half-width. Second, binning issues further complicate the selection of optimal LEDs for building the cluster. The effect of a limited and uncertain selection of peak wavelengths is illustrated by optimizing two trichromatic and two tetrachromatic clusters, each composed of commercially available high-power LEDs (R,G,B,A and warm white), for colour quality and luminous efficacy (Fig. 3a ). Two sets of red, green, blue, amber and warm white LEDs were obtained from two different suppliers. As can be seen from Fig. 3b, the red, green and amber LEDs from the two suppliers had almost identical spectral power distributions. However, the blue and warm white phosphor LEDs showed a substantial difference. The spectral power distributions of the LED clusters were again optimized using the NSGA-II genetic algorithm. Because the peak wavelengths and spectral widths were fixed for real LEDs, the optimization problem was solely characterized by the fluxes of the individual LEDs.
It is clear from Fig. 3a that changing from a set of modelled LEDs to a set of real LEDs, with fixed peak wavelengths and FWHMs, had a dramatic effect on the obtainable luminous efficacy and colour quality as measured by the S a metric. Binning was also found to have a potentially large effect, as can be clearly seen from the differences between the Pareto fronts of LED clusters composed of LEDs from two different suppliers (dots versus circles in Fig. 3a). For example, colour quality differences as large as 0.08 and 0.05 S a-units, were respectively observed for the RGB and RGBA clusters. However, as can be seen in Fig. 3b, the differences in spectral power distribution between the two sets of LEDs were only substantial in the case of the blue LED (∆λpeak = 17 nm).
A comparison of the Pareto optimal front of the different types of LED clusters showed that real tetrachromatic clusters performed better than trichromatic clusters even though the simulated RGB clusters were no worse than any of the other simulated cluster types. The tetrachromatic phosphor type LED was found to have a potentially higher colour quality (as estimated by the Sa-metric) than those containing an amber LED. However, the latter were able to achieve a much greater luminous efficacy, but only at lower colour quality. Tetrachromatic LED clusters of the R/G/B/phLED type were therefore found to have the most potential to design a solid-state white light source with good colour quality (S a ≥ 0.8) and reasonable luminous efficacy of radiation (250-300 lm/W).
Finally, as is clear from Fig. 3a, all optimized real LED clusters had a higher luminous efficacy and potentially higher colour quality than the CIE reference illuminants. The superior colour quality of a LED lamp optimized for the colour quality metric was confirmed by the authors in a psychophysical experiment . In a paired comparison experiment with 92 lay observers, the colour quality of an optimized 2700 K tetrachromatic cluster (S a = 0.790; CIE Ra = 81) was found to be significantly better than the colour quality of a 2700 K halogen light source (S a = 0.766; CIE Ra = 100). Note that the Sa-metric was indeed able to correctly predict the better perceived colour quality of the LED cluster, while CIE Ra could not due to its fundamental problem of combining non-optimal reference illuminants with a strict colour difference metric.
The spectral power distributions of simulated and real tri- and tetrachromatic LED clusters were succesfully optimized for luminous efficacy of radiation and colour quality as measured by the memory colour quality metric by calculating the Pareto optimal solutions with the multi-objective NSGA-II genetic algorithm. Investigation of the subset of Pareto optimal solutions with a colour quality higher than that of the CIE reference illuminants revealed optimum peak wavelengths and spectral widths of the LEDs in each cluster type. It was found that almost all of the optimum peak wavelengths were located close to either the peak wavelength of the CIE 1924 luminous efficiency function or to one of Thornton’s three prime colours. The Pareto optimal fronts of real LED clusters were always located at smaller luminous efficacy values and colour quality than those of the simulated LED clusters. Binning was found to have a potentially substantial impact on the achievable colour quality of real LED cluster. Of the four types of LED clusters investigated real tetrachromatic LED clusters composed of an RGB LED and a warm white phosphor were found to have the greatest potential for designing a LED cluster with good luminous efficacy and colour quality. No or only negligible differences were found between the different types of simulated LED clusters. LED clusters could achieve a better colour quality (based on memory colours) and higher luminous efficacy than a set of CIE reference illuminants. This was confirmed by the authors  in a psychophysical experiment with 92 observers, where a 2700 K tetrachromatic LED cluster (S a = 0.790; CIE Ra = 81) was found to have a better perceived colour quality than a halogen light source (S a = 0.766; CIE Ra = 100).
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