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Abstract
The color gamut volume (CGV) and light efficiency of a four-primary display system were theoretically simulated with different wavelength configuration. Given the wavelengths of the blue and red primaries, we optimized the other two primary colors; the wavelength set with the largest CGV was chosen. The maximum CGV, 2.346 × 106, was obtained at (660, 530, 507, 465) nm. The maximum light efficiency was also determined. A trade-off between CGV and light efficiency should be made according to the requirement of the devices. This study provides guidance for the construction of a four-primary laser display system and the optimization of the CGV in multi-primary display systems.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser display [1] is a representative technology of a wide color gamut display [2], which has a large color gamut, even surpassing ultra-high definition (UHD) standard Rec. 2020 [3]. Color gamut [1,4] is considered to be among the most important parameters of a display system, representing the color rendering capacity.
To further widen the color gamut, many researchers have focused on multi-primary displays (MPDs) [5–7]. Many studies have used a broad spectrum light for four-primary [5,10,11], five-primary [12], and even six-primary [13] systems. Moreover, some important research was conducted on narrow spectrum light source such as quantum dot [14], from three-primary to multi-primary [15,16], and achieved a color gamut as large as 118.60% Rec. 2020 [17]. The signification and possibility of color gamut enhancement were proved in principle.
In 2019, Wang et al. [18] proposed a theory for fast and accurate calculation of color gamut volume (CGV) for a three-primary laser display system; this method was applicable to any spectrum distribution, including discontinuous or very peaked spectra. A general theory of MPDs compatible with laser displays was established by Yao et.al [19]; based on this, further studies were conducted on four-primary laser display systems to enhance the color gamut in a special color region (e.g., yellow or green), and the gamut coverages were also studied [20,21].
Although progress has been made in six-primary laser display systems [22], four-primary systems are closer to commercial application and production owing to the complexity and cost of extra lasers and color channels. A key question is how the CGV of four-primary laser display systems is affected by the selection of the wavelength, and which wavelength sets could be adapted in the future to maximize the CGV.
Buckley [23] studied the optimal wavelength in a three-primary system, limited to a two-dimensional (2D) CIE-xy diagram. Liu [24] determined the optimal gamut for MPDs; however, this was also limited to the 2D CIE-xy or u’v’ diagram. Owing to a lack of luminance properties, these 2D diagrams are not sufficiently accurate. Rodríguez-Pardo [25] presented a method to select the optimal primary in a display system. He derived a spectral model of the system that could be applied to different display technologies, and a non-linear constrained maximization of the gamut volume. However, the choice of the blue and red wavelengths yielded low visual efficiency that led to low electric-optical efficiency, resulting in a larger CGV. Additionally, this study did not refer to the evolving regular CGVs, which is an important aspect.
In addition to the CGV, the luminous flux of a system is also an important parameter in the choice of the wavelength. The common designation is “luminous efficacy” or “luminous efficiency.” The term “luminous efficacy” is used by the CIE to characterize the luminous flux per watt of electrical energy input (lm/W). The term “luminous efficiency” in the CIE definition means the number of lumens per optical power of radiation (also in lm/W) [26]. The former considers the laser electric-to-optical power conversion efficacy more than the latter, which depends on the specific lasers. Therefore, “Efficiency” is more familiar to many. Luminous efficiency was typically adopted in the study of luminous optimization, whereas luminous efficacy was used to evaluate the energy conversion of the actual display device. Several related studies were conducted on three-primary LED [27], micro-LED [28] and laser [23], liquid crystal four-primary [29], and light-emitting diode four-primary [30] systems. However, research on the optimization of the light efficiency in four-primary laser display systems is still lacking. Such topic is necessary and worth studying.
In this study, the CGV and light efficiency were theoretically simulated at different wavelength configuration of a four-primary display system. Further analysis and discussions were also performed.
2. Theory basis of four-primary laser display systems
We performed simulations in the widely used uniform CIELAB 3D color space. However, since our calculations convert CIEXYZ to CIELAB, this approach can also be easily converted to other spaces, including CIELUS, CIECAM02, CIECAM16, and even Jzazbz.
According to Yao et al. [19] and based on the sample in our previous paper [20], using the method of representing the fourth primary by the initial three primaries, all possible luminance ratios of four-primary systems can be determined. Furthermore, the CGV of any four-primary display system can be calculated. The process is as follows.
Given the white point, e.g., D65 (x = 0.3127, y = 0.3290), Yr, Yg and Yb are set as the luminances of the initial three-primary system. A constant luminance of the fourth primary, Yfourth, can be determined by the luminances of the initial three primaries, where Yr’, Yg’ and Yb’ denote the corresponding luminances of the initial three primaries:
The final luminance of the primary colors are denoted as YR, YG, YB and YFOURTH in the four-primary system. We set YFOURTH =kYfourth, where k is the co-efficiency of the fourth primary and the only variable. The luminance of each primary color in practical display systems cannot be negative; hence, we have the following constraints:
Each k that satisfies Eq. (2) corresponds to a unique set of four-primary luminance ratios. The range of k is [0,1]. Applying the algorithm, the CGVs of all wavelength sets for each luminance ratio can be determined.
With the algorithm, the light efficiency of a four-primary system can also be calculated using the luminance ratio in Eq. (2). Ignoring the electric conversion efficiency of the laser and assuming a total luminance of 100 lm, the optical light power of each primary can be determined as
3. Theoretical simulations of color gamut volume
Figure 1 shows the spectrum visual efficiency curve of human eyes, which represent the bright vision at luminance more than 3 cd/m2 and predominates the normal displays. Several typical values are listed in Table 1. The bright visual efficiencies for wavelengths of 380-780 nm in Table 1 show that the optical light power demanded is significantly higher for wavelengths shorter than 445 nm or longer than 680 nm, despite the larger CGV.
Table 1. Bright visual Efficiency for several sample wavelengths [31]
To simplify the simulation in a multi-primary system, the settle wavelengths of blue and red lasers were selected here. Noting that the wavelength choice may not be the optimal, it did not, however, affect the simulations and results.
Blue laser sources with a 445 nm primary are widely available and have a low cost. However, the use of a shorter blue wavelength excludes a significant part of the NTSC color standard [23], as shown in Fig. 2. The blue triangle with 445 nm does not include the left-down part of the NTSC standard, whereas the red triangle with 465 nm includes it, as marked with yellow circle.
With the popularization of display devices, people turn to pursue the comfort of viewing. Excessive optical radiation represented by blue light hazards causes damage to the human body [32,33], as shown in Fig. 3. The blue light hazards can be divided into two types: visual and non-visual hazards [34]. Visual harm refers to the damage to retinal cones and rods caused by blue light irradiation, which is mainly generated at 400–500 nm [35] and reaches a maximum at 435 to 440 nm [36]. Non-visual hazards refer to the effect of blue light on the secretion of melatonin when the retina is exposed. Related research [37–39] showed that the inhibition curve of the human melatonin secretion stimulated by blue light presented a Gaussian distribution with a central wavelength of 488 nm. However, the non-visual hazards somehow help reinforce our circadian rhythm. The problem occurs only when we are ready to sleep. Therefore, the long-wavelength blue laser around 488 nm does not affect the usage of a 465 nm laser in display systems.
Additionally, the visual efficiency at 445 nm is 0.0298, whereas the value at 465 nm is 0.0739, which is more than double that at 445 nm.
Therefore, for comprehensive consideration of the lack of gamut coverage with NTSC standard, visual hazards of short blue light, and visual efficiency, a longer wavelength of 465 nm was chosen as the fixed blue primary.
Considering the electric-optical efficiency, a wavelength of 660 nm was chosen as the fixed red primary. However, 660 nm is only a representative value; if 670 nm or other wavelengths were adopted, similar conclusions would be obtained.
3.1 Single-variable simulation for a three-primary laser display system
In this section, we consider the white point to be set at D65. To simplify the calculation, the range of the wavelength of the third primary, λ3, can be limited using the CIE x-y diagram, shown in Fig. 4; W is the white point D65, and A1 and A2 are the intersections of the extension of the RW and BW lines with the boundary of the horseshoe, respectively. If the third primary is between B and A1 or A2 and R, the white point W will be outside the triangle and the three-primary system will not be balanced with a positive tristimulus value; instead, it should be between λ30 and λ31. The coordinates of A1 and A2 are (0.0317, 0.3631) and (0.4441, 0.5547), corresponding to the wavelengths 493 and 570 nm, respectively. Hence, the range of the third primary is [493, 570].
The black curve in Fig. 5 shows the CGVs for different values of the third primary in the range [493, 570] at intervals of 1 nm. The CGV increases quickly and reaches a maximum of 2135000 for a wavelength of 519 nm, before rapidly decreasing. Hence, the optimal wavelength set for a three-primary laser display system is taken to be (660, 519, 465) nm.
Fig. 5. CGVs for different wavelengths of the third primary at 1 nm intervals in the range [493, 570].
The red curve in Fig. 5 shows the light efficiency of a three-primary system. With an increase in λ3, the light efficiency increase monotonically. The maximum value 376.06 lm/W is obtained at the wavelength set (660, 570, 465) nm. The light efficiency is 112.43 lm/W when the CGV reaches the maximum at λ3 = 519 nm. A trade-off can be made between the demand on the CGV and light efficiency.
3.2 Simplified single-variable simulation in a four-primary laser display system
3.2.1 Simulation of CGV
In Section 3.1, we outlined the decision to set the wavelength of the third primary color to be 519 nm. We will now determine the optimal wavelength of the fourth primary, λ4, which lies in the range [466, 659]. We evaluated the CGVs using the method outlined in Section 2 for the wavelength set (660, λ4, 519, 465) nm, where the fourth primary was varied with an interval of 1 nm from 466 to 659 nm. The results are shown in Fig. 6, where the x-axis is the wavelength of the fourth primary, the y-axis is the luminance coefficient k (100×k is the luminance of the fourth primary), and the z-axis is the CGV. Two peaks occur around 501 nm (k = 0.20) and 541 nm (k = 0.27), with CGVs of 2.297 × 106 and 2.257 × 106, respectively. The two peaks are separated by a gap; as the wavelength of the fourth primary increases, the CGV decreases until 519 nm and then increases until 541 nm. For the wavelength set (660, λ4, 519, 465), when λ4 increases from 493 to 519 nm, the k corresponding to the maximum CGV increases from 0.03 to 0.35; similarly, when λ4 increases from 519 to 659 nm, the k corresponding to the maximum CGV decreases from 0.35 to 0.07. Figure 7 shows several example of CGVs at λ4 = 480, 501, 520, 540, 560, 570, 580, 600, 620, and 640 nm for varying k. Clearly, the maximum CGV is obtained for λ4 = 501 nm; the CGV increases when λ4 increases from 466 to 501 nm, then decreases when λ4 increases from 501 to 659 nm; the range of k increases to 0.94 when λ4 increases from 466 to 570 nm, then decreases when λ4 increases from 570 to 659 nm.
Fig. 6. CGVs for the wavelength set (660, 519, 465) nm for different wavelengths of the fourth primary.
Fig. 7. Sample CGVs for the wavelength set (660, λ4, 519, 465) nm for different k values for λ4 = 480, 501, 520, 540, 560, 570, 580, 600, 620, and 640 nm.
3.2.2 Simulation of light efficiency
Similar simulation was performed for the light efficiency parameter. The light efficiency of each possible wavelength set (660, λ4, 519, 465), with λ4 varying in the range [466, 659], is shown in Fig. 8(a). Here, for each wavelength set, the light efficiency is the maximum value with varying luminance ratio k.
Fig. 8. Light efficiency variation (a) with λ4 for the wavelength set (660, λ4, 519, 465) nm and (b) with the luminance ratio k for the wavelength set (660, 571, 519, 465) nm.
Figure 8(a) shows that the light efficiency first decreases from 101.44 to 77.28 lm/W (minimum value at 493 nm), and then increases rapidly. The maximum value 414.06 lm/W is obtained for λ4 = 571 nm. Then, it decreases with increasing λ4.
Because the visual efficiency at 571 nm is larger than that at the other three primaries, the light efficiency at λ4 = 571 nm increases monotonically with an increase in its luminance ratio k.
3.3 Double-variable simulation in a four primary display system
3.3.1 Simulation of CGV
Section 3.2 details a special situation, in which the wavelength of the third primary is predefined. Here, we set both the third and fourth primaries as two variables to be determined.
As in Sections 3.1 and 3.2, the wavelengths of the red and blue primary were set to 660 and 465 nm, respectively. The third primary is denoted as λ3 and the fourth primary is denoted as λ4. We performed a double-loop traverse of the wavelength set (660, λ4, λ3, 465) nm, where λ3 was in the range [493, 570] and λ4 was in the range [466, 659]; both were varied with an interval of 1 nm.
This was a very computationally intensive simulation. To reduce the computational cost, the third and fourth primaries were first varied with an interval of 5 nm to obtain a general distribution of the CGV. Then, we repeated the simulation with an interval of 1 nm for smaller ranges of λ3 and λ4.
Figure 9 shows the results of the double-variable simulation with a 5 nm interval; the CGV is larger for λ3 and λ4 in the ranges [493, 519] nm and [520, 570] nm, respectively. From the direction of the vertical x axis (λ3), for a fixed λ3, with λ4 varying from 466 nm to 660 nm, the CGV increases rapidly to the maximum, and then decreases to a stable level. The stable range here is from 570 to 660 nm; because the color coordinates on the color boundary of this segment fits x + y≈1, z≈0, the CGV changes little. This is close to the interpretation in the 2D color diagram. From the direction of the vertical y axis (λ4), the CGV at a fixed λ4 increases first and then decreases with λ3.
Fig. 9. CGVs for the wavelength set (660, λ4, λ3, 465) nm for different wavelengths of the third and fourth primaries, with an interval of 5 nm.
We then varied λ3 and λ4 in their ranges with an interval of 1 nm; the results are shown in Fig. 10. The maximum CGV of 2346226 was obtained for λ3 = 507 nm and λ4 = 530 nm; hence, the wavelength set (660, 530, 507, 465) nm yields the largest CGV. This result is similar to that of Rodríguez-Pardo [21]. The CGV was within 5% of the peak when the third and fourth primaries were in the ranges [493, 519] and [520, 553], respectively.
Fig. 10. CGVs for the wavelength set (660, λ4, λ3, 465) nm for different wavelengths of the third and fourth primaries, with an interval of 1 nm.
To make it clearer, the CGV at the wavelength set with nine single λ4 (493, 501, 507, 520, 530, 540, 550, 560, and 570 nm) and λ3 belonging to [466, 659] nm was plotted in Fig. 11, where the wavelength of the third primary color increases from 501 to 507, 520, 530, 540, 550, and 560 nm; the colorful curves represent the CGV with the corresponding λ3 and all possible values of λ4. The two peaks of CGV are clearer. The maximum CGV 2346226 was obtained for λ3 = 507 nm and λ4 = 530 nm (green curve).
We further investigate the performance of this optimal wavelength set. The variables in the algorithm in Section 2 have the following values:
The luminance values of the four primary colors are given by
The range of k is 0 ≤ k ≤ 0.71. Hence, the CGVs can be obtained using the algorithm and are shown in Fig. 12. The maximum CGV is 2346226, obtained for k = 0.39; the corresponding luminance is 26.5703:39:29.8735:4.5562. The gamut coverage of Rec. 2020 can be determined in a similar manner and is also shown in Fig. 9. The gamut coverage for Rec. 2020 for different values of k is also shown in Fig. 10; the maximum coverage is obtained for k = 0.67, which corresponds to the luminance ratio 29.0716:67:3.8997:5.0287. We note that the largest gamut coverage for any gamut standard can be simulated by the same method.
Fig. 12. CGV and gamut coverage of Rec. 2020 for different values of k for the wavelength set (660, 530, 507, 465) nm.
The maximum CGV is approximately 126.50% of the Rec. 2020 gamut coverage; the gamut coverage of Rec. 2020 is approximately 91.69%. The CGVs for different luminance ratios are detailed in Section 2. Figure 13(a) shows the corresponding scheme in a CIE-xy diagram, while Fig. 13(b) shows the maximum color solid in CIELAB space.
Fig. 13. Scheme for the wavelength set (660, 530, 507, 465) nm in (a) CIE-xy diagram and (b) maximum CGV in CIELAB space.
To obtain an intuitive expression of the color gamut in different regions, we adopted the method of gamut rings proposed by Masaoka [40,41]. The gamut rings in polar coordinates [22] are displayed in Fig. 14, where B = [(a*RSS)2 + (b*RSS)2]1/2 represents the distance between a point and the black point in CIELAB space; the area enclosed by the drawn curve is the gamut volume; a*RSS and b*RSS are defined by (a*RSS, b*RSS)L*, h = (C*RSS(L*, h) cos(h), C*RSS(L*, h)sin(h)) [40,41]; and h is the hue angle. The arrows represent the hue angle directions of the marked colors. Compared with the gamut standard of Rec. 2020, the CGV of the four-primary wavelength set with k = 0.39 covers well the area of Rec. 2020, missing only approximately 8.30% in the yellow region; this demonstrates excellent compatibility with the UHD display standard. The four-primary CGV yields enhancements in the green and cyan regions, which could be used as an expanded reserve for future wider gamut standards or personalized customizations. Figure 15 shows the gamut rings for the wavelength set (660, 530, 507, 465) nm for different k values, and hence different luminance ratios. As k increased, the CGV decreased in the cyan and turquoise regions but increased in the dark green and yellow regions. This is because the luminance ratio for a fourth primary wavelength of 530 nm increases as k increases, while that for a wavelength of 507 nm decreases.
Fig. 15. Gamut rings of the wavelength set (660, 530, 507, 465) nm for different values of the luminance coefficient k.
3.3.2 Simulation of light efficiency
With similar simulation, the light efficiency in a four-primary system with two variables was obtained and is shown in Fig. 16.
Fig. 16. Light efficiency of a four-primary system: sample optical light efficiency for the wavelength set (660, λ4, λ3, 465) nm for λ3 = 493, 501, 507, 520, 530, 540, 550, 560, and 570 nm.
In Fig. 16, the x-axis is the wavelength of the fourth primary, the y-axis is the optical light efficiency, and the colorful curves represent the wavelength of third primary. For a fixed λ3 except 570 nm, with λ4 increasing from 493 to 659 nm, the optical efficiency in most situations decreases first, reaches the minimum value (about 80 lm/W) at λ4 near 493 nm, then increase to the maximum value (about 400 lm/W) at λ4 near 571 nm, and decreases monotonically afterwards. For a fixed λ4 except 570 nm, with λ3 increasing from 493 to 560 nm, the variation range of the optical efficiency is reduced. The red curve represents the light efficiency of the wavelength set (660, λ4, 501, 465) nm, and the maximum value 489.96 lm/W obtained at (660, 571, 501, 465) nm is also the maximum among all wavelength sets.
4. Discussion
With the constraint of the white point set at D65, the maximum CGV was found to be 2346226 for the wavelength set (660, 530, 507, 465) nm at the luminance coefficient k = 0.39. Compared with the three-primary system, which has a maximum CGV of 2135000, the increase is approximately 10%. Considering the large base, the gamut expansion is considerable. The most obvious advantage is the enhancements in the green and cyan regions. However, this enhancement is not fully appreciated in the cyan region due to the low proportion of cyan objects in nature.
Moreover, with the constraint of the white point set at D65, the maximum light efficiency 489.96 lm/W was found at the wavelength set (660, 571, 501, 465) nm. Compared with the three-primary system, which has a maximum light efficiency of 376.06 lm/W at the wavelength set (660, 570, 465) nm, the increase is approximately 30%.
However, it is impossible to determine an optimal wavelength set of the four-primary system, because the maximum value of CGV and light efficiency were not obtained at the same wavelength set and same luminance ratio k. It should be selected by our preferences.
As shown in Fig. 17(a), for the wavelength set (660, 530, 507, 465) nm with the largest CGV at k = 0.39, the optical efficiency is 113.67 lm/W. In this wavelength set, if we want a relatively balanced performance, a larger k could be adopted.
Fig. 17. CGV and light efficiency of the wavelength sets (a) (660, 530, 507, 465) nm and (b) (660, 571, 501, 465) nm.
As shown in Fig. 17(b), for the wavelength set (660, 571, 501, 465) nm with the largest light efficiency, the CGV is rather low with a large k value, which is clearly meaningless. The best method is to choose the k between 0.4 and 0.6.
With the above procedure, the CGV and light efficiency of any four-primary wavelength set could be calculated and compared. A trade-off should be made between these two parameters. This allows manufactures to customize their products according to the visual characteristics and preferences of each customer, by choosing a different fourth primary and its luminance ratio k, which is unfeasible in a three-primary system.
5. Conclusion
In this study, the CGV and light efficiency were theoretically simulated at a different wavelength configuration of a four-primary display system. Considering the improvement of electric-optical efficiency and reduction of blue light hazards, we predefined the wavelengths of the blue and red primaries, and then varied the other two primary colors; by considering the evolution of the CGV, the wavelength set that yielded the largest CGV was determined. A maximum CGV of 2346226 was obtained for a wavelength set of (660, 530, 507, 465) nm. The gamut distribution and evolution of the wavelength set (660, 530, 507, 465) nm were further investigated. The light efficiency performance was also studied and discussed. A trade-off should be made between these two parameters according to the demands. This study provides clear guidance for the construction of a four-primary laser display system and the personal optimization of multi-primary display systems.
Funding
National Key Research and Development Program of China (2021YFF0307804).
Disclosures
The authors declare no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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