Taking color quality scale (CQS) as color rendering assessment criterion, the parameters including each color LED’s peak wavelength λi and fractional radiant flux Ii are optimized using genetic algorithm to maximize the luminous efficacy of radiation (LER) of the spectral power distributions (SPDs) of multi-color white light source with 3 to 7 components while maintaining 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 SPDs is analyzed. It is shown that to achieve a Qa greater than 95 (5-color LEDs) or even close to 100 (7-color LEDs), the spectral energy could be concentrated in the range of 410~675 nm, indicating that this wavelength range makes a major contribution to high color rendering properties. Spectra filtering experiments show that spectrum around 580nm is harmful to color rendering. To obtain a white light source composed of 3-color LEDs with CQS Qa ≥ 80 and correlated color temperature (CCT) within 2700-6500K, the energy ratios among 410-495nm, 495-595nm, and 595-675nm intervals, can be simplified as that of the reference source with the same CCT.
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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 .
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 . Therefore, the white light source by mixture of multi-color LEDs  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 . 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 . 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 λi, full width at half maximum (FWHM) Δi and fractional radiant flux Ii 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 , full width at half maximum (FWHM) and fractional radiant fluxes Ii. According to commercialized LED products, such as Phlips Lumileds Lighting Luxeon Rebel family, the values of c and 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
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 as7, 8]. M is the steps of MacAdam ellipse, and 7-step MacAdam ellipse is used here in accord with provisions of ANSI C78.377 , while 4-step or other step MacAdam ellipse could also be used . In this paper, the solid state lighting source with CCT of #### K means that the chromaticity coordinates of this source stay within the 7-step MacAdam ellipse centered at coordinates of the reference source with CCT of #### K.
The constraint on color rendering properties of the solid state lighting source is specified by CQS , recently introduced for rating color rendering properties of solid-state lamps, expressed as5], and Qc is a given CQS value.
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 λi, Δi and Ii (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  could not be used, and the optimization problem becomes a nonlinear optimization problem with nonlinear constraints. As Δi 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 (λi, Ii, 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 . 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 , 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 .
3. Results and discussion
3.1 Optimized SPDs for white LED source with different number of color LEDs
Firstly, the maximum Qa that white LED source can achieve for different CCTs using different number of color LEDs, is investigated. This is obtained by maximizing Qa(, , Ii, i = 1,2,…,N) under constraints of Eq. (6) and Eqs. (1)–(4). The results show that for CCT of 2700~6500K the maximum Qa 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 Qc 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.
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 Qa 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 Qa 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 , but is close to Einhorns’ observation that the useful radiation is located in the 430-660nm range .
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 Qa of about 60 and CCT of 3500K and that after modification are shown in Fig. 3 . Then the change of Qa 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 Qa is lower. The improvement of Qa 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 . 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 Qa reduces from 100 to about 84.
In the previous report of Thornton , 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 Qa > 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 Qa 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 Qa 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.
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 Qa equal to or greater than 80 with that of reference source.
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 Ii of the B-, G-, R-LED should be close to these energy ratio, can be used to reduce the search domain of Ii in Eq. (6). For example, the bond of Ii could be set as energy ratio*0.7 <Ii <energy ratio*1.3 instead of Eq. (6).
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 Qa 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.
This work was supported by the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2011BAE01B07, and 2012BAE01B03), the National Basic Research Program of China (Grant Nos. 2011CB301902, and 2011CB301903), the High Technology Research and Development Program of China (Grant Nos. 2011AA03A112, 2011AA03A106, and 2011AA03A105), the National Natural Science Foundation of China (Grant Nos. 61176015, 60723002, 61176059, 60977022, and 51002085).
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