Ultra-Wide Color Gamut of Three-Primary and Four-Primary Laser-Based Displays With Large Circadian Tunability

A numerical and optimization study has been well performed on both the circadian effect and color gamut of laser-based displays (LBDs) consisted of multiple-primary laser diodes (LDs) to realize the ultra-wide color gamut. The LBDs mainly consist of red (R), green (G), and blue (B) LDs (namely by 3-LD scheme), and also consist of red (R), yellow (Y), green (G), and blue (B) LDs (namely by 4-LD scheme). The Gaussian function is adopted to simulate the spectra of LDs due to their extremely narrow full-width at half maximum (FWHM < 2 nm). In order to evaluate the circadian effect of such LBDs, the circadian parameters, such as melanopic efficacy of luminous radiation (MELR), circadian action factor (CAF), and circadian stimulus (CS), are considered and employed in this investigation. The genetic algorithm (GA) is used to search for the optimal values of circadian parameters at a certain thresholds of color gamut. The Rec. 2020 standard is used to evaluate the color gamut, and 70%, 80%, 90%, 95%, and 100% Rec. 2020 are five selected thresholds. The optimal results are across the correlated color temperatures (CCTs) from 2700 K to 6500 K. The circadian tunablity is deduced by the ratio of the maximum MELR at 6500 K CCT to the minimum MELR at 2700 K CCT. The circadian performances are compared with the light-emitting diode (LED)-based displays. It is believed that this work could provide a useful guidance for the circadian regulation of people by the multiple-primary laser-based displays.


I. INTRODUCTION
U LTRA-WIDE laser-based displays (LBDs) can be achieved by using three or four laser diode (LD) spectra [1], [2], [3], [4]. Compared with the displays based on lightemitting diodes (LEDs), they possess relatively narrower fullwidth at half maxima (FWHMs) that are generally within 0. nm, which is quite beneficial for the realization of wider color gamut than LED-based displays because the color coordinates of LDs are nearly located on the spectral locus in the Commission Internationale de L'Eclairage (CIE) chromaticity diagram. Importantly, to the best of our knowledge, the LDs are also efficiency-droop-free lighting devices, whereas these so-called efficiency-droop phenomena usually occur in the GaN-based LEDs [5], [6]. Therefore, the LBD based on multiple LDs may become an alternative promising display technology in addition to the LED-based displays, organic LED (OLED) displays, or liquid crystal displays (LCDs) [7].
Since the first discovery of intrinsically photosensitive retinal ganglion cells (ipRGCs), more and more researches have been concentrated on the design of healthy lighting for the applications of general lighting or displays [8], [9], [10], [11], [12], [13], [14], [15]. Most of published research studies are on the healthy lighting from multiple-primary LEDs, a few of them are on the multiple-primary LBDs. This fact urges us to carry out a comprehensive investigation on the healthy LBDs. Previously, our research groups have performed a study on the circadian and color gamut of three-primary LBDs [16], where the color gamut is mainly evaluated by Rec. 2020 [17]. Here, we comprehensively investigate the ultra-wide color gamut and high circadian tunabilities of LBDs with three and four primary colors, and derive a relationship between the color gamut and optimal circadian performances based on the optimization tools designed by the genetic algorithm (GA). A comparison between LBDs and LED-based displays is also done in this work. Finally, we believe that this work could provide a useful reference for the possible utilization of LBDs on the circadian regulation of people in the daily life.

A. Spectral Modelling of LBDs
As well-known to all, the LBDs can consist of red (R), green (G), and blue (B) LDs (namely by 3-LD scheme), also consist of red (R), yellow (Y), green (G), and blue (B) LDs (namely by 4-LD scheme). The former is generally called as three-primary LBDs, while the latter can be called as four-primary LBDs. Due to the extremely narrow FWHMs, the LD spectrum truly follows a Gaussian shape. Thus, for simplicity, we adopt the Gaussian function to model LD spectrum in the LBDs. Here, the Gaussian function is described as [18], where λ, λ 0 , and Δλ are wavelength, peak wavelength, and FWHM, respectively; i is the primary number. Thus, according to above expression, the spectral power distribution (SPD) of LBDs contains N primary colors can be expressed as, where a i is the relative intensity of primary color, and it is located in the range of (0, 1). Increasing the primary number of LBDs would increase the device complexity and the cost. Therefore, the number of primary colors is selected as N = 3 and N = 4 for simplicity.

B. Color Gamut and Circadian Performances
In order to perform this investigation, the melanopic efficacy of luminous radiation (MELR), circadian action factor (CAF), and circadian stimulus (CS), related to the SPD of LBDs are generally used to evaluate the circadian performances of lighting. Here, we mainly adopt MELR as the circadian parameter during the optimizing process, while the other two parameters, CAF and CS, are as a supplement to the MELR.
Among the parameters that for circadian performance, the circadian action factor (CAF) is generally defined as [12], where C(λ) is the normalized spectral sensitivity function to describe the circadian action (as seen in Fig. 1); V(λ) is the normalized CIE photopic spectral sensitivity function (as seen in Fig. 1); S(λ) is the SPD of white light sources.
As specified by the standard of CIE S 026/E:2018 [19], the MELR (K MELR ) is written as, where M(λ) is the normalized spectral sensitivity of ipRGC photoreceptors to optical radiation incident at the cornea (Fig. 1); The circadian stimulus (CS) proposed by Rea et al. is expressed by [10], where CL a is called as the circadian light. The color gamut is very important for the colorful display. A new color triangle was devised previously to strictly define the color gamut standard, which has been called as Rec. 2020. Zhu et al. previously demonstrated a quantum-dot (QD) based micro-display with over 80% of color gamut of Rec. 2020 [17]. The Rec. 2020 gamut range is the widest among all gamut standards at present. While comparing the triangular area of display (A display ) with the triangular area of Rec. 2020 (A standard ), the color gamut coverage ratio is generally adopted to describe the color gamut (G c ), which is written as [17], In this work, we adopt the color gamut coverage ratio instead of the color gamut ratio for the description of color gamut.

C. Non-Linear Program Based on Genetic Algorithm
The non-linear program (NLP) based on the common genetic algorithm (GA) [20] is designed for solving this optimization issue. This study is performed under eight correlated color temperatures (CCTs, 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K, and 6500 K, respectively). This non-linear program is designed on the programming platform of Matlab. Fig. 2 briefly shows the flow chart of this work. First, we generate 3-LD model and 4-LD model based on the Gaussian function. Then, we determine the ranges of wavelengths for the 3-LD scheme and 4-LD scheme. The objective function is proposed as Max. MELR and Min. MELR. Subsequently, we set penalty functions such as CCT ranges, CCT deviation, and color difference (D uv ) in the CIE 1960 UCS chromaticity diagram. The related parameters of LBDs including the wavelength ranges, CCT ranges, CCT deviation, and color difference (D uv ) are listed in Table I in detail. In addition, the FWHM of each primary color in the LBDs is set as 0.1∼2 nm due to the narrow spectra of laser. After running the GA-based NLP, the optimal results are output to perform related analysis.

III. RESULTS AND DISCUSSION
A. Optimal Results of Three-LD Scheme Fig. 3(a) and (b) show the optimal results for three-LD scheme mainly in terms of MELR. We have calculated the maximum and minimum values of MELR under eight typical CCTs, that are 2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K, and 6500 K, respectively. From this figure, one can see that the MELR value is increasing with increasing CCT. Also, the MELR is increasing with increasing threshold of Rec. 2020, for obtaining the minimum MELR, whereas the MELR is decreasing with increasing threshold of Rec. 2020, for obtaining the maximum MELR. The luminous efficacy of radiation (LER) is also an important parameter of light. The LER values corresponding to low MELR and high MELR are plotted in Fig. 3(c) and (d). The LER for low MELR case is found as 256 ∼ 384 lm/W, while 86 ∼ 282 lm/W for high MELR case.
In addition, we calculate the circadian tunability which is evaluated by the ratio of Max. MELR at the CCT of 6500 K to Min. MELR at the CCT of 2700 K, which can be expressed The calculated R values for different Rec. 2020 thresholds is clearly plotted in Fig. 4. When the threshold of Rec. 2020 is increasing, the ratio R describing the circadian tunability is decreasing monotonously. Clearly observed from this curve, it is not a regularly linear trend. Therefore, we adopt the quadratic polynomial function to fit this curve which describes the relationship of Rec. 2020 versus R, and the related coefficients in the quadratic polynomial function are also shown in Fig. 4, as well as an excellent coefficient of determination R-square value (R 2 ) of 0.9998. This empirical function can be conveniently utilized to predict the circadian tunability under various thresholds of Rec. 2020 within the ranges from 70% to 100% for three-LD scheme, thus dispensing with much labour.

B. Optimal Results of Four-LD Scheme
Fig. 5 plots the optimal results of four-LD scheme. The MELR is also increasing with CCT, which is similar to that of three-LD scheme. Also, the Rec. 2020 threshold is set as 70%, 80%, 90%, 95%, and 100%, respectively. Interesting, we can clearly observe that, for the four-LD scheme, after the threshold of Rec. 2020 is larger than 90%, the MELR does not change or changes slightly. This phenomenon is quite different from that of three-LD scheme. For 4-LD scheme, the LER values corresponding to low MELR and high MELR are plotted in Fig. 5(c) and (d). The LER for low MELR case is found as 314 ∼ 467 lm/W, while 213 ∼ 365 lm/W for high MELR case. These LER values are higher than 3-LD scheme.
In Fig. 6, we can note the ratio of MELR for evaluating the circadian tunability for four-LD scheme for five thresholds of Rec. 2020 (70%, 80%, 90%, 95%, and 100%, respectively). The curve can also be fitted by the quadratic polynomial function and good R-square value (R 2 = 0.9938) can be achieved. While the Rec. 2020 > 90%, the circadian tunability of R is nearly Fig. 5. For the four-LD scheme, (a) low and (b) high values of MELR under eight typical CCTs (2700 K, 3000 K, 3500 K, 4000 K, 4500 K, 5000 K, 5700 K, and 6500 K) while Rec. 2020 threshold is set as 70%, 80%, 90%, 95%, and 100%, respectively. The corresponding values of LER can be found in (c) and (d). Fig. 6. Ratio of MELR for evaluating the circadian tunability for four-LD scheme for five thresholds of Rec. 2020 (70%, 80%, 90%, 95%, and 100%, respectively). The curve can be fitted by the quadratic polynomial function and good R-square value (R 2 = 0.9938) can be achieved.
unchanged. This expression can be used to predict the circadian tunability of MELR for four-LD scheme. The CCT, CAF, MELR, circadian tunability, and the optimal wavelength combination for a specific Rec. 2020 (90%) for different schemes (3-LD scheme and 4-LD scheme) are listed in Table II. At the 2700 K CCT, we have the minimum MELR, while for 6500 K, we have the maximum one. The deduced value of R is then 4.42 for three-LD scheme, which is about twofold higher than that of four-LD scheme (R = 2.25) at the same color gamut. It indicates that the three-LD display shows higher circadian tunability than four-LD display. Previously, we have reported a work on the circadian performances of LED-based displays [21]. We have calculated the optimal circadian tunability based on the MELR as 3.95 for the same Rec. 2020 of 90%. The comparison of circadian tunability among 3-LD, 4-LD, and 3-LED scheme is also plotted in Fig. 7. Thus, we can conclude that the 3-LD scheme has shown the highest circadian tunability while  compared with 4-LD scheme or LED-based display (3-LED scheme). Due to the narrower FWHM of LD in LBDs compared with that of LED, both MELR and CAF are lower at 2700 K and higher at 6500 K under the same color gamut, thus, 3-LD has higher circadian tunability than 3-LED. This is in consistent with the statement "it is very important to select the FWHMs of LEDs as narrower as possible for obtaining larger circadian tunability as much as possible" in the previous work [21].
In addition, the optimal wavelength combination for three-LD scheme and four-LD scheme is also listed in Table II. The given wavelength combinations here can become a useful reference for the realization of LD-based display with high circadian tunability in the future.

C. Analysis of Circadian Stimulus
The circadian stimulus (CS), circadian light (CL a ), and corneal illuminance (Lux) for three-LD and four-LD schemes are also calculated. In addition to the influence of the spectra of LD combination, the circadian effects are also greatly affected by the illuminance or brightness of light sources in the display. Rea et al. have previously found that a light exposure level at eyes with a CS > 0.3 or greater is effective in stimulating the circadian system of people [22]. Previously, we have also discussed the CS for three-LED scheme [21]. But more cases have been shown here, including more different CCTs, different LD schemes, and different optimization goals. Fig. 7 shows the Lux of 3-LD and 4-LD scheme versus CCT while CS value is fixed as 0.1 and 0.3, respectively, while the color gamut Rec. 2020 is fixed as 90%.
Prior to the analysis, we make a statement that, the light with CS < 0.1 is more helpful for sleep at night, while light with CS > 0.3 is more helpful for increasing the working efficiency in the daytime. Fig. 8(a) shows Lux versus CCT for 3-LD scheme, while the core SPD is related to minimum MELR in Fig. 3(a). Fig. 8(b) illustrates the Lux versus CCT for 3-LD scheme while the SPD is corresponding to the maximum MELR ( Fig. 3(b)). Fig. 8(c) and (d) are Lux versus CCT for 4-LD scheme with minimum MELR (Fig. 5(a)) and maximum MELR (Fig. 5(b)), respectively. The CS has increased with increasing corneal illuminance. We can see that the corneal illuminance is the highest at 4000 K CCT among eight CCT values while CS = 0.1 and 0.3. At the same level of corneal illuminance, the SPD with 4000 K CCT has lowest CS values among eight CCTs. That is, 4000 K light source can provide the smallest CS, thus having most potentials for improving sleep quality. At the same level of CS, the 4000 K light source requires more illuminance than other CCTs (for example, for 3-LD scheme in Fig. 8(a), in order to let the light source own a CS = 0.1, 4000 K light source requires about 200 lx, while others require less than 200 lx). In the daytime, light with CS > 0.3 may increase the working performance. Thus, the light with 6500 K CCT has more potentials over other CCTs in terms of energy saving. This fact can be explained as that in the daytime, the 6500 K light source requires least illuminance for the same level of CS = 0.3. Particularly to say that, the maximum values happen at 3500 K CCT (not at 4000 K CCT) for 4-LD scheme while the SPD is related to the maximum MELR in Fig. 8(d).

IV. CONCLUSION
In summary, we perform a numerical investigation on the spectral optimization of circadian effect and color gamut for three-primary and four-primary laser-based displays. Eight CCT values from 2700 K to 6500 K are considered in this work. The common genetic algorithm is used as the effective optimization tools for searching for the optimal values on circadian effects. The circadian effects are evaluated by the melanopic efficacy of luminous radiation (MELR), circadian action factor (CAF), and circadian stimulus (CS). For the display, the circadian values are increasing with the increment of CCT. This phenomenon is quite in consistence with the general lighting case. At the same time, as the thresholds of Rec. 2020 are increasing, the varying trends of MELR are opposite to each other while acquiring the maximum and minimum values. The value of R is 4.42 for three-LD scheme, which is about twofold higher than that of four-LD scheme at the same color gamut Rec. 2020 of 90%. After a comparison of 3-LD scheme (R = 4.42), 4-LD scheme (R = 2.25), and 3-LED scheme (R = 3.95), we can conclude that 3-LD scheme possesses the highest circadian tunability.