Study on the correlations between color rendering indices and the spectral power distribution

The intrinsic spectrally resolved sensitivity (ISRS) of color rendering indices (CRIs) is investigated by using spectral loss simulations. It is demonstrated that Ra exhibits large sensitivities around 444, 480, 564, and 622 nm, while for R9 the sensitivity peaks are around 461, 581 and 630 nm, which all shift slightly with the correlated color temperature. If considering the ISRS as a bridge between the spectral power distribution of LED and its CRI, one could obtain a high CRI by minimizing the deviation between the shapes of the illuminant spectrum and the reference spectrum, both after modulations by the ISRS as a weighting function. This approach, recommended as a guideline for the spectra design aiming at a high CRI, is described and justified in depth via a mathematical model. This method is spectra-oriented and could largely facilitate the spectra design. ©2014 Optical Society of America OCIS codes: (230.3670) Light-emitting diodes; (330.0330) Vision, color, and visual optics; (330.1710) Color, measurement; (330.1715) Color, rendering and metamerism; (330.1730) Colorimetry. References and links 1. E. F. Schubert and J. K. Kim, “Solid-state light sources getting smart,” Science 308(5726), 1274–1278 (2005). 2. J. M. Phillips, M. E. Coltrin, M. H. Crawford, A. J. Fischer, M. R. Krames, R. Mueller-Mach, G. O. Mueller, Y. Ohno, L. E. S. Rohwer, J. A. Simmons, and J. Y. Tsao, “Research challenges to ultra-efficient inorganic solidstate lighting,” Laser & Photon. Rev. 1(4), 307–333 (2007). 3. Y. Shirasaki, G. J. Supran, M. G. Bawendi, and V. Bulović, “Emergence of colloidal quantum-dot light-emitting technologies,” Nat. Photonics 7(1), 13–23 (2012). 4. T. Erdem, S. Nizamoglu, X. W. Sun, and H. V. 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Introduction
Since the beginning of last decade, solid-state lighting has been developed at an exciting speed as a new generation of illuminants [1].Due to its various advantages, such as high luminous efficiency and tiny size, etc., a growing number of commercial lamps made of lightemitting diodes (LEDs) appear on the shelves of supermarkets and websites of online stores, gradually encroaching the market share of incandescent and florescent lamps.However, before them could totally replace conventional illuminants, LEDs still face many research challenges, among which, the design for the spectral power distribution (SPD) is a critical one, for the SPD mainly determines the property of a illuminant [2].Different from conventional light sources, the SPD of a LED is the mixture of peaks emitted from chips and phosphors, which facilitate us to cast it by changing the quantities or types of luminous components.The colloidal quantum-dots LED developed recently is even more adept at such SPD tailoring since it contains a series of narrow-band emissions peak at different wavelengths from nanoparticles with various diameters [3].All of these innovations give rise to the significance of the SPD design [4][5][6][7][8][9][10][11][12][13][14].Generally, there are 3 main parameters, i.e., luminous efficacy of optical radiation (LER), correlated color temperature (CCT), and color rendering index (CRI, or R a ), which are set to gauge the chromaticity qualities of spectra [5].The ultimate goal of SPD design is to obtain a SPD with both high LER and CRI under a given CCT [6][7][8].It is nonetheless always not easy to satisfy this requirement because of the well-known trade-off between the LER and the CRI [5].The former is a quasi-subjective factor since its calculation involves the luminosity function, which peaks at 555 nm in the photopic vision and at ~498 nm in the scotopic vision.Hence, one could obtain the SPD with a high LER by making the intensity as high as possible at the vicinity of 555 nm (suppose in photopic vision) and otherwise as low as possible.Unfortunately, there exists no such a simple rule on improving the CRI, which is obtained by quantifying the deviation of color appearance when a light source under test and a reference source shining on a series of samples respectively.In fact, it is of more significance to enhance the CRI than the LER in practice, since the total efficiency of an illuminant is not only decided by the LER but also by the external quantum efficiency of both chips and phosphors.In the literature, despite of some recent reports about generating white light by a series of laser sources [15,16], it is widely accepted that a continuous SPD across the visible region without any significant loss in intensity is a basic requirement on generating a high CRI [6,7,10,13].In particular, the SPDs of LEDs, either the type of blue chip exciting yellow phosphors or the type of multichips, have spectral losses, which have been considered as the main reason of the low color rendering property.However, it still lacks of clear criteria that could describe the internal relationships between the SPD and the CRI, because of its highly abstract calculating procedure.Unlike the LER, the CRI itself hardly can provide a direct link to the SPD.In consequence, most of the works reported about SPD designs were based on the scarce empirical guidelines accumulated beforehand or by picking the one with the highest CRI among a larger amount of SPDs generated randomly via computers, among which the He's research group has contributed much thorough work on the SPD design [8,9,13,14].On the other hand, although so far being considered as the most systematic parameter indicating the color rendering quality, the CRI has subjected controversies, which mainly focus on the insufficiency for only averaging 8 samples in the final index.It gives rise to the attention of R 9 , which represents the color rendering property for the deep red [5][6][7][8][9][10][11][12][13][14].In addition, some other methods for evaluating the color rendering property have been explored, such as color quality scale (CQS), reported by NIST [17], and the new approach presented by Žukauskas, by calculating the percentage of the number of correctly rendered samples out of a total of 1269 samples of different colors [12].
According to the experiences from the SPD design for LEDs, in this article, we aim at revealing the correlations between the CRI and the SPD, and trying to provide a detailed guideline in the SPD design for the purpose of obtaining a high CRI.Note that the SPDs used as reference illuminants always has R a of 100, and intensity losses on any wavelengths would distort the color rendering property to various extents.It is worth conducting comparisons between simulated SPDs (s-SPDs) and reference SPDs (r-SPDs) under the same CCT, in order to discover R a and R 9 sensitivity in different spectral regions.However, few reports have focused on the detailed color rendering sensitivity upon wavelengths.Through a series of theoretical spectral loss simulations, we discovered that CRIs are extremely sensitive in some selective spectral regions whereas are less or even little sensitive otherwise, which could be well described as the intrinsic spectrally resolved sensitivity (ISRS) for each CCT.
We then consequently employed it as a modulation function and established a mathematical model showing high R a , as well as high R 9 , R 11 and R 12 , could be obtained by making the modulated shape of s-SPD as close to that of r-SPD as possible.This model resembles a mathematical transformation that turns the vague relationship between CRI and SPD into a visualized and easily comprehensible one.Compared with those solutions involving Monte Carlo method, it is spectral-oriented thus would greatly enhance the efficiency of R&D.

Spectral loss simulation
We presented a series of simulations in order to measure the spectrally resolved CRI sensitivity.The simulations were carried out on the software named "Chromaticity", as introduced in Ref [18], and Microsoft Excel 2010 with VBA programing.As sketched in Fig. 1, a well was drilled in the r-SPD, by setting the value of a specific spectral region to zero (in Fig. 1, the well-width is 50 nm).Those modified r-SPDs have a lower CRI than that of the original ones.By swiping the well over the whole visible spectral region step-by-step, and measuring the CRIs of modified r-SPDs with wells in each different positions, we were able to obtain the influence of spectral loss on the CRI.The testing was carried out on three r-SPDs on 3500 K, 5455 K, and 8500 K with a well-width set on 50 nm.(Specifically, the one of 5455 K is considered to have a comparatively uniform intensity over the visible spectral range, for its CCT is identical with that of "equal-energy spectrum".)The results of R a and R 1 -R 9 are illustrated on Fig. 2(a)-2(c), as well as the r-SPDs in each CCT on Fig. 2(d)-2(f) respectively.Thick solid lines highlight the data of R a and R 9 .Almost all of the CRIs have been distorted around 555 nm, showing deep valleys in the yellow region, especially for the moderate and high CCTs.In addition, a majority of them have 2 shallow valleys alongside the deepest ones on the red and blue region respectively.On each end of the visible region near 380 and 780 nm, all of the CRIs show little distortions.Among all of the CRIs values, the R 9 is the most vulnerable one, experiencing distortions more than 30 in most of the wavelengths, leading to extremely low values which are even minus within the yellow region.In addition, dissimilar to others, the distortion patterns of R 9 in all 3 CCTs have a deep valley in the red region.Among the 3 CCTs, the distortion patterns exhibit some deviations.Taking R 9 for example, in 3500 K, the valleys with blue and green region seem not as deep as they are in 5455 K and 8500 K.With the help of the r-SPDs in the related CCTs, shown on Fig. 2(d)-2(f) respectively, one could figure out that the depth of the valleys is changing with the r-SPDs: the distortion rate tends to be strong in the spectral region where r-SPDs are intensive, and vice versa.To investigate on the influence of the well-width, we repeated the simulation using different well-width of 5, 10, 20, and 50 nm respectively and the results of R a and R 9 are illustrated in Fig. 3, in which only the results of 5455 K are shown.As the well-width increases, the distortions in both R a and R 9 are becoming severe, while the patterns keep roughly unchanged, with only the small valleys in the blue and red regions gradually merging into the main green one in R a .

CRI sensitivity
The results of spectral loss simulation inspired us to investigate into the spectrally resolved CRI sensitivity.As has mentioned previously, losses in different spectral regions lead to various distortions in CRI, which are CCT dependent and vary among different indices.Since SPD designers need to take into account the improvement of the LER, R a and R 9 simultaneously under a given scenario, it would be significant to discover the rules, follow which the CRIs reach the theoretical ceiling.It could be approached with the help of the hints given by the spectral loss simulations.We interpreted the spectrally resolved distortion rate plotted in Fig. 2 and Fig. 3 as a telltale of the CRIs sensitivity, i. e., the larger distortion, and the higher sensitivity.For example, on the 5455 K, the huge decrease of R a in the green region suggests a high sensitivity around 550 nm.In order to gain more subtle plots for these sensitivities, we performed the same simulations as we did previously but limited the wellwidth to 1 nm, and process the results using the Eq. ( 1) Equation ( 1) is a linear transformation from the spectrally resolved distortion  The ISRS of R a and R 9 exhibit large differences to each other under the same CCT.Although we use arbitrary units herein, the intensities of each ISRS curves could still indicate the sensitivities among 4 CCTs for both R a and R 9 , hence, R 9 has the magnitude of sensitivity 4-5 times larger than that of R a .In such detailed plots, the ISRS of R a generally exhibit a 4peak structure with each peaks at around 444, 480, 564, and 622 nm respectively, whereas, for R 9 , 3 peaks with each around 461, 581 and 630 nm.It suggests that the optical intensity should be high within those peak areas and any spectral loss in such regions would cause severe CRI distortion.For R a , with the increasing CCT, the intensities of 3 peaks below 600 nm keep roughly the same ratio to each other, while the one around 630 nm is growing up.A similar phenomenon could be observed in R 9 : The red peak is raising its intensity compared to the other two, which means that when designing LEDs with a high CCT, a proper red light composition around 630 nm need to be mixed into the whole spectrum.These large dissimilarities between the sensitivity patterns of R a and R 9 reveal the huge difference in requirements on the SPD when optimizing these two factors and that is the reason why there are always SPDs with decent R a , but minus R 9 .
The algorithm of CRIs requires the reflectivity spectra of each test color sample for calculating the color deviations between the r-SPDs and r-SPD with spectral loss.The ISRSs proposed herein are derived from CRIs with the help of r-SPD.Hence, the ISRS could be regard as the equivalents to the reflectivity spectra in mathematics.Whereas, after the reflectivity spectra being transformed into the ISRSs, the correlation between the SPDs and the CRIs is uncovered.

Guideline for the CRI optimization
Since the obscure characteristics of CRI, there is a lack of a theoretical guideline that describes how to give a light source a good color rendering ability.Although one can inefficiently iterate such process including immediately calculating the CRI data, modifying the SPD accordingly, and checking its CRI again and again until it reaches an acceptable value, it requires a more intelligent method which could link the CRI to the shape of SPD, thus paving the way towards the solution of a high CRI.The ISRS could serve as such a bridge between the SPD and the CRI, since it to some degree uncovers the rules that lead to high respective CRIs.Nonetheless, ISRS is still inadequate to be a guideline until we have been able to establish a mathematical model and yield a factor correlating with the shape of SPD and its CRI simultaneously.Following the sketchy discussion in previous sections, we presented a thorough model.
As indicated in the ISRS, for a specific CRI, it shows larger sensitivity in some spectral regions whereas less in others.Therefore, considering the ISRS as the weighting function, we compared the shapes of s-SPD and r-SPD both after the modulation by ISRS, and measured the modulated deviation (MD, or d MD ) as a single combined factor.Since the ISRS shifts with the CCT, as discussed in previous sections, we only employed the moderate ones of R a and R 9 on 5455 K as the weighting function, for simplicity.Equations ( 2) give the process of MD determination as follows.
The first two equations in Eqs.(2) represent the modulation on both the s-SPD [I(λ)] and r-SPD [R(λ)] which are to be normalized to have the same area beneath the curves, restricting the following comparison only between their shapes, i.e., ( ) M I λ and ( ) M R λ .Noting that the deviation introduced by the spectral white noise or spectral spikes have little influence on the CRI but huge on the MD, we need to smooth the SPD curves to eliminate these high frequency deviations.A two-level summation is employed as described in the third equation of Eqs.(2).The spectral regions are divided into 3 and 4 sub-regions for the calculation of R 9 and R a , by the minimum of each ISRS, as shown in Fig. 4, respectively.The inner operator  n sums the deviations between ( ) M I λ and ( ) M R λ on each wavelength within each sub-region, aiming to reduce the unwanted noise.The outer operator sums the absolute values over all N sub-regions (for R a and R 9 , N = 4 and N = 3 respectively).Using this algorithm, we tested MDs and CRIs of a plenty of simulated LEDs spectra which consist of several Gaussian functions [18], by continuously changing the parameters of the Gaussian peaks such as intensity, full width at half maximum, etc.We figured out the general relationship between the CRI and the MD, i.e., the higher the CRI, the lower the MD.
The plots of R a and R 9 vs. MD of the simulation on a set of 3-peak spectra are illustrated in Fig. 5(a) and 5(b) in black solid squares respectively, and the parameters of these spectra are listed in Table 1.Note that those color points with the distances to the blackbody locus larger than 0.05 in CIE 1960 UCS color diagram have been abandoned, since they are unsuitable for general lighting.With the rising intensity of the third peak from 0 to 300, the R a increases to ~86 then decreases, as its MD exhibits an inverse trend.The trend of R 9 is more precise, for the increasing and decreasing parts are almost overlapped.This reveals a high coherence between MDs and its related CRIs.The black dashed lines in Fig. 5(c) shows the evolution of the s-SPDs with the increase in intensity of the third peak, and the red solid line highlights the s-SPD with the highest R 9 = 95.5 and R a = 84.9(on the intensity of the third peak of 91).What would it turn out to be if one just makes the comparison between the s-SPD and the r-SPD without any modulation?To answer this question, we carried out the same simulation with the absence of the weighting function, by setting the ( )  because when the R a falls down, the un-modulated deviation losses its obedience as it decreases too, which is similar for R 9 .In stark contrast, the MD shows a good coherence with the R a or R 9 over both the rising and decreasing parts.We also extended this algorithm to R 11 and R 12 , which represent the color rendering ability for strong green and strong blue samples, respectively.As did for the R 9 , we vary the intensities of green and blue peaks to check the correlations between R 11 and R 12 with their respective MDs.The parameters of the 3 Gaussian peaks are listed in Table 1.Since the blue and green peaks simulate the emissions from the chips and phosphors respectively, it is unrealistic to change their intensities severely in the simulation as for R 9 , which mimics the intensity changing in the red chip emission.Figure 6 illustrates the two simulations and the related SPD changing.Similar to the plots in Fig. 5  The calculation process of MDs is much simpler than that of CRIs.Additionally, unlike the CRI, the MD is directly derived from the SPD but closely related to the CRI.Having these two advantages, the MD could be considered as an intermediary that fills the huge gap between SPD and CRI.With the help of the MD, one could tailor the s-SPD to make it close to r-SPD after modulation, and when the MD reaches its minimum, the CRI would be fairly high.In accordance with this guideline, we simulated a set of SPDs with 4 peaks on a series of specific wavelengths and with fixed FWHMs, by adjusting the intensity of each peak to make both MDs as small as possible, which, in a diagrammatical manner, makes the two modulated s-SPD and r-SPD alike to each other in terms of R a , R 9 , R 11 and R 12 respectively.If turning the MD is still something abstract, making two plots overlap to each other is much  2.The related obtained s-SPD with the r-SPD is plotted on Fig. 7(a).Their modulated versions for R a , R 9 , R 11 , and R 12 are on Fig. 7(b)-7(e) respectively.In all four forms of modulations, the s-SPD and r-SPD lie very close to each other, without huge difference as spotted in the original ones on Fig. 7(a).During the evaluation, we accumulated some tips for generating high-CRI SPDs.That is, choosing the emissions that peak at the vicinity of the ISRS peaks to improve the overlap ratio, especially in some subtle structures such as those within 450~550 nm in Fig. 7(d).Such a little refinement could largely improve the CRI.Note that some works reported recently employ lasers diodes to generate the white light with a high R a .The SPDs of lasers consist of several spikes with extremely narrow linewidths, which seems contradict to our theory [15,16].However, the accuracy of the CRI of those SPDs containing narrow peaks has been under debate [19], and the whole lighting industry tends to use light sources with smooth SPDs for the consideration of human health.In a word, it is safer to use a light source of which the SPD is closer to those of natural ones.

Summary
In this work, we investigated into the relationship between the CRI and the SPD, and discovered the wavelength dependence sensitivity of CRI.By using the spectral-loss simulation, we could obtain such sensitivity and plot it out as the ISRS.In advance, noticing the ISRS could be considered as a bridge between the CRI and the SPD, we developed a theoretical model and calculate the modulated shape difference between s-SPD and r-SPD.The MD has a high coherence with the CRI, which reveals that the CRIs depend largely on the difference between the modulated shape of s-SPD and r-SPD.Hence, benefiting from this, we recommended a guideline for the SPD design in order to gain high CRIs, which is to minimize the deviation the s-SPD and r-SPD under the modulation of the ISRS.This is a simple and SPD-oriented method that could largely avoid the blindness when designing the LEDs spectra.Despite of its high accuracy, the model presented herein is after all a basic one.Therefore, in the future, more works are still needed to study the characteristic of the ISRS of each specific CRI in detail, and to uncover more information about the CRI.

Fig. 1 .
Fig.1.The sketch of the spectral loss method.The well swipe over the whole visible spectral region from 380 to 780 nm, as the well-width could be adjusted.

Fig. 2 .
Fig. 2. The CRI distortions in the spectral loss simulation with well-width set on 50 nm, on (a) 3500 K, (b) 5455 K, and (c) 8500 K respectively, and the related r-SPDs on (d) 3500 K, (e) 5455 K, and (f) 8500 K respectively.Solid lines highlight R a (black) and R 9 (red) respectively, and other indices are illustrated as dashed lines.

Fig. 3 .
Fig. 3.The (a) R a and (b) R 9 distortions on a changing well-width, which ranges from 5 to 50 nm.
respectively) divided by the related normalized r-SPD ( ) R λ in order to eliminate the influence of un-uniformity of r-SPD intensity, and forming the intrinsic spectrally resolved sensitivities [ISRS, or ( ) i η λ ] for R a and R 9 respectively.In the following sections, we will present a series of simulations to check its accuracy.Here we first give some results for R a and R 9 in different CCTs along with the normalized r-SPD in each CCT respectively, which are illustrated on Fig.4.

.
un-modulated deviation.The results plotted in Fig. 5(a) and 5(b) along with the MD curves clearly indicate the necessity of employing the ISRS as the weighting function,

Fig. 5 .
Fig. 5.The plots of MD (black solid square) and unmodulated deviation (red empty square) versus (a) R a and versus (b) R 9 .(c) The evolution of the s-SPD, in which the one with the highest R 9 highlighted in red solid line.

Fig. 6 .
Fig. 6.(a) The plot of MD with Ra, R 11 and R 12 in the simulation of changing the green peak, and the related SPDs are in (b).(c) The plot of MD with R a , R 11 and R 12 in the simulation of changing the blue peak, and the related SPDs are in (d).
(a) and 5(b), the MD of each CRIs deceases monotonically as CRI increases, indicating the reverse correlation as expected.

Fig. 7 .
Fig. 7. (a) The obtained s-SPD in the simulation with R a = 94.4,R 9 = 97.2,R 11 = 90.7 and R 12 = 88.3 and the related r-SPD; their modulated versions for (b) R a , (c) R 9 , (d) R 11 and (e) R 12 .Note that when the CRIs reach high values, the modulated s-SPD and r-SPD appear very alike.

Table 2 .
The Gaussian parameters of the SPD in Fig.7