1. Introduction

It is the aim of lighting technology to achieve a comfortable lighting environment with energy consumption as low as possible. In addition to the energy saving property which has been usually stressed when we talk about light sources, for general daylight illumination applications they should have high colorimetric performance, such as appropriate apparent color and good color-rendering capabilities [1].

Solid state light source based on LEDs is on its way to large-scale applications in general lighting, and is believed to someday replace traditional lighting sources, such as incandescent lamps and fluorescent tubes. The current widely applied approach for commercial white light generation by LEDs is a combination of yellow phosphor with a blue LED. It has been successfully applied in outdoor lighting, such as street lighting, due to relative high efficiency compared with traditional lighting source. However, it is difficult to achieve high color rendering properties due to the lack of long-wavelength component in spectrum. As a result, it can’t be widely used in indoor daylight illumination applications, e.g. illumination in museums, homes, offices, and stores [2]. Therefore, the white light source by mixture of multi-color LEDs [2] has been widely studied. However, the relationship between colorimetric properties and the SPDs of multi-color white LED sources has not been systematically investigated.

The white light source with multi-color LEDs has a narrow-band spectra feature, which makes it inadequate to assess color rendering properties with the widely used CRI (color rendering index), recommended by the CIE in 1965 [3]. CIE 177 has given the comment that current CRI based ranking of a set of light sources containing white LED light sources contradicts the visual ranking [4]. To improve the color rendering assessment, some modifications are recently made on the CRI and several new figures of merit are proposed, among which CQS attracts more and more attention as it overcomes some of the shortcomings of CRI while maintaining its advantages [5, 6].

In this paper, CQS is taken as the color rendering assessment criterion, and the SPD of white light source with multi-color LEDs is described as a function of the peak wavelength λ_i, full width at half maximum (FWHM) Δ_i and fractional radiant flux I_i of each color LED. These parameters are optimized using genetic algorithm to maximize LER of the SPD while keeping the deviation of its color and color-rendering capability from that of the reference light source within the specified scope. Then the wavelength dependence of these optimized SPDs is analyzed.

2. Physical model and mathematical procedure

White LED spectra studied here are composed of three or more narrow bands corresponding to the emission of multi-color LEDs. The emission line of the i-th color LED is approximated by Gaussian shapes with peak wavelength ${\lambda }_i$, full width at half maximum (FWHM) ${\Delta }_i$ and fractional radiant fluxes I_i. According to commercialized LED products, such as Phlips Lumileds Lighting Luxeon Rebel family, the values of c ${\lambda }_i$ and ${\Delta }_i$ are

Then SPDs are optimized for the maximum luminous efficacy under given color and color rendering properties. The flowchart of establishing the optimization model is shown in Fig. 1 . The objective function is the LER, defined as

 

Fig. 1 The flowchart of establishing the optimization model.

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The CIE recommends black-body radiation or CIE daylight as reference source which is thought to have excellent colorimetric performance at a given CCT. The color and color rendering properties of white LED source is described by the deviation from that of the reference source. In this paper, we focus on establishing white LED sources that have the same color as a CIE reference source and proximate color rendering capability.

The constraint on the color of white LED source is specified by the deviation from that of the reference source, i.e. black-body radiation or CIE daylight for CCT below and above 5000K, respectively. The MacAdam ellipse [7, 8] centered at the chromaticity coordinates of the reference source is used to define this constraint, expressed as

The constraint on color rendering properties of the solid state lighting source is specified by CQS [5], recently introduced for rating color rendering properties of solid-state lamps, expressed as

The optimized SPD of the solid state lighting source is obtained by solving the optimization problem that maximizing the LER in Eq. (5) under the constraints given in Eqs. (4), (6) and (7) with λ_i, Δ_i and I_i (i = 1,2,…,N) as variables, and with bounds specified in Eqs. (1)–(3). Because of the constraint on the light source color expressed in Eq. (2), the three common color-mixing equations [11] could not be used, and the optimization problem becomes a nonlinear optimization problem with nonlinear constraints. As Δ_i have been prior determined, this optimization problem can be solved by searching inside the 2N- or (2N-1)- (taking Eq. (7) into account) dimensional parametric space of peak wavelengths and fractional radiant fluxes (λ_i, I_i, i = 1,2,…,N). Classic optimization techniques usually do not work well in finding the global optimized point. The optimization method used in this paper is the genetic algorithm (GA), which is a stochastic global search and optimization method that mimics the process of natural selection and genetics of Darwin's theory of biological evolution [12]. It repeatedly modifies a population of individual solutions through operations of selection, crossover and mutation, and the population evolves to the global optimized solution. It makes no requirements about the searching space [13], and is suitable for the optimization problem of white LED source. Furthermore, in order to make it more effective patternsearch is used as a hybrid function [14].

3. Results and discussion

3.1 Optimized SPDs for white LED source with different number of color LEDs

Firstly, the maximum Q_a that white LED source can achieve for different CCTs using different number of color LEDs, is investigated. This is obtained by maximizing Q_a(${\lambda }_i$, ${\Delta }_i$, I_i, i = 1,2,…,N) under constraints of Eq. (6) and Eqs. (1)–(4). The results show that for CCT of 2700~6500K the maximum Q_a of white LED source with 3, 4, 5, and 7 colors are in the range of 80~85, 90~95, 95~100, and approximate 100, respectively.

Given Q_c as 80, 90, 95, 95 for 3, 4, 5 and 7 colors white LED source in Eq. (7) and CCT as 3500, 5000 and 6500K, the optimized white LED spectra are obtained, as shown in Fig. 2 . The LER of these SPDs are also shown in Fig. 2.

 

Fig. 2 Optimized spectra of 3-, 4-, 5- and 7- colors white LED source with near maximum Q_a for CCT of 3500K, 5000K, and 6500K.

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3.2 SPDs variation with different number of color LEDs

As the number of colors increases, the spectrum gradually extends to the short-wavelength and long-wavelength at the same time. As a result, the spectrum occupies a larger wavelength range, and meanwhile the power distribution becomes more uniform in the entire wavelength range. To achieve a Q_a greater than 95 (5-color LEDs) or even close to 100 (7-color LEDs), the spectral energy can be concentrated in the range of 410-675 nm. This indicates that, if Q_a is taken as the color rendering metric, spectrum in wavelength range of 410~675nm makes a major contribution to high color rendering properties while the role of spectrum in the short-wavelength (<410 nm) and the long-wavelength (> 675nm) range is less important. This is different from Walter’s observation that emission lower than 440nm or higher than 620nm is wasted [15], but is close to Einhorns’ observation that the useful radiation is located in the 430-660nm range [16].

3.3 Harmful wavelength to color rendering properties of multi-color white LED source

It is worth noting that intensity around 580nm always remains low, indicating that this wavelength range has very small contribution to or even is harmful to color rendering, which is consistent with previous reports based on CRI [17, 18]. To check this effect, spectra around 580nm are removed from the original spectral by setting the intensity of spectrum in 570-590nm to zero. As an example, the original optimized SPD of 4 color-LEDs with Q_a of about 60 and CCT of 3500K and that after modification are shown in Fig. 3 . Then the change of Q_a after modification is investigated. The results are shown in Table 1 and Table 2 for 3- and 4-color LEDs, respectively. It is shown that the color rendering properties can be effectively improved by removing spectrum of this region, especially when Q_a is lower. The improvement of Q_a of 4-color LEDs by spectral filtering is greater than that of 3-color LEDs. These results indicate that spectrum around 580nm is harmful to color rendering. This is in accord with Thornton’s report [17, 18] but different from Walter’s observation [15]. As a comparison, spectra filtering experiments are also conducted for 3500K and 5000K blackbody radiator and 6500K CIE daylight. After removing the spectrum of 570-590nm, the CQS Q_a reduces from 100 to about 84.

 

Fig. 3 The modification of optimized SPDs. Taking the original optimized SPD of 4 color-LEDs with Q_a of about 60 and CCT of 3500K and that after modification as an example.

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Table 1. Change of Q_a After Removing Spectrum Around 580nm for 3-color LEDs

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Table 2. Change of Q_a After Removing Spectrum Around 580 nm for 4-color LEDs

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In the previous report of Thornton [17], the wavelength range around 495nm is also found to be disastrous for the CRI, but we do not find this effect in our case for CQS. Removing wavelength range around 495nm does not result in significant improvement of CQS, even makes it worse.

3.4 Wavelength dependence of optimized SPDs for 3-color white LED source

As described in section 3.2, the more number of colors the white light source has, the higher color-rendering properties it achieves. However, the difficulty of color control and the cost increase along with number of colors. White light source with 3-color LEDs is a good trade off that it could get very acceptable color-rendering properties (CQS Q_a > 80) as well as easy color adjustment. So, in this section a further analysis of the wavelength dependence of optimized SPDs of 3-color white LED source is carried out. As mentioned above, the maximum Q_a of 3-color LEDs white spectra is in the range of 80~85. For CCT of 2700~6500K, the SPDs of white light source with 3-color LEDs are optimized for the maximum luminous efficacy with CQS Q_a equal to or greater than 80, and the results are shown in Fig. 4 . Obviously, peak wavelengths of 3-color LEDs are in the red, green and blue region respectively, and they are labeled as R-, G- and B- LED respectively. Center wavelengths of R-, G- and B-LED under different CCT are shown in Table 3 . As described in Section 2, a white LED source with CCT of #### K means not that it has the same chromaticity coordinates with the reference source, but that its chromaticity coordinates stay within the 7-step MacAdam ellipse centered at coordinates of the reference source with CCT of #### K. This makes the variation of SPDs of white LED source with CCT much complicated. But by and large, as can be seen from Fig. 4, In addition to the individual SPDs, as CCT increases from 2700 to 6500K, the center wavelength of R-, G- and B-LED all shift to shorter wavelength, the R-LED’s intensity continuously reduces and the Blue LED's intensity increases, while the G-LED’s intensity remains nearly constant.

 

Fig. 4 SPDs of 3-color LEDs white source optimized for the maximum luminous efficacy under CCT of 2700K~6500K and CQS Q_a equal to or greater than 80.

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Table 3. Center Wavelengths of R-, G- and B-LED Under Different CCT

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In Fig. 5 , for CCT of 2700K~6500K, we compared SPDs of 3-color LEDs white source optimized for the maximum luminous efficacy with CQS Q_a equal to or greater than 80 with that of reference source.

 

Fig. 5 Compare of SPDs of 3-color LEDs white source optimized for the maximum luminous efficacy under CCT of 2700K~6500K and CQS Q_a equal to or greater than 80 with SPDs of respective reference source.

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As the color temperature increases, the energy gradually shifts to the short wavelength region, while energy ratio of the mid wavelength region changes lightly. We find that the wavelength range of 410-675nm could be divided into three intervals of 380-495nm, 495-595nm, and 595-675nm, which are labeled B-, G-, R-intervals, respectively. The energy ratio of the B-,G-,and R-interval in the optimized SPDs of 3-color white LED source and that in the reference source, are approximately the same, as shown in Fig. 6 . This finding could be used to simplify the optimization problem of 3-color white LED source. For example, to get the optimized SPDs of 3-color white LED source with CCT of ####K, firstly the SPD of the reference source with CCT of ####K is calculated, and then the energy ratio of the B-,G-,and R-intervals in the reference source can be obtained. The observation that fractional radiant flux I_i of the B-, G-, R-LED should be close to these energy ratio, can be used to reduce the search domain of I_i in Eq. (6). For example, the bond of I_i could be set as energy ratio*0.7 <I_i <energy ratio*1.3 instead of Eq. (6).

 

Fig. 6 Energy ratio between B-, G- and R-interval of optimized SPDs of 3-color LEDs white source and that of the reference source with the same CCT.

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4. Conclusion

White LED spectra with 3 to 7 components are optimized for the maximum luminous efficacy under given CCT and given color rendering properties. The results indicate that, spectrum in 410-675nm wavelength range makes a major contribution to high color rendering properties, while the role of spectrum in the short-wavelength (<410 nm) and the long-wavelength (> 675nm) range is less important. Spectra filtering results shown that spectrum around 580nm is harmful to color rendering, and the color rendering properties can be effectively improved by removing spectrum in this region, especially when Q_a is lower. For optimized SPDs of 3-color LEDs white source, it is found that the wavelength range could be divided into three intervals of 380-495nm, 495-595nm, and 595-675nm, respectively, and the energy ratio of these intervals is approximately the same as that of the reference source. It should be pointed out that the same analysis has been also carried out using CRI Ra as the color rendering assessment criterion, and the results also hold if CRI Ra is used.