All-reflective RGB LED flashlight design for effective color mixing

Hsiang-Chen Wang; Yao-Ting Chiang; Chung-Yi Lin; Ming-Yen Lu; Min Kai Lee; Shih-Wei Feng; Chie-Tong Kuo

doi:10.1364/OE.24.004411

1. Introduction

Given its high energy efficiency, light-emitting diode (LED) has been widely used in various applications, such as in scene illumination, displays, and traffic lights. Hence, LED plays an important role in daily life. Scene illumination is the most common color-mixing application. When used for scene illumination, multi-colored LEDs are mixed to render a color cast illumination effect that is suitable for a special occasion with the help of optical elements, such as diffusers, light pipes, and batteries of lenses. In a restaurant, the light used for scene illumination can be applied to usher customers into different stages of an activity and to control on-site atmosphere. In a boutique, product value can be improved by a scene created by light. Red, green, and blue (RGB) LED has a narrow spectrum and low color rendering index; however, its major advantage of having a narrow-band light source lies in its capability to improve visual contrast [1], and thus, it can help viewers identify specific objects. Hence, RGB LED can be an optimal light source for design, as already proven in previous literature [2–4].

In 2011, Sun et al. introduced an effective mixing method for RGB LEDs. They used a color-mixing module that consisted of a circular light pipe, a diffuser, and total internal reflection (TIR) lenses. The light pipe mainly assisted in importing light, whereas the diffuser was used to improve the color-mixing effect. Finally, the TIR lenses were used to diffuse light from a Lambertian light source. Through this color-mixing optical framework, the main color-mixing module improved color uniformity with the help of the diffuser [5]. In the optical framework proposed by Julius et al., a parabolic reflector was used to generate parallel light rays and reflect them. A micro-lens array was then adopted to achieve the color-mixing goal. Using a micro-lens array was advantageous because it produced a good color-mixing effect even if beam divergence was small; finally, the light shape of the color mixture produced was a soft and uniform circular light [6]. Liu et al. proposed a LED-based color-mixing method that could provide high color uniformity and uniform irradiance. Two independent types of rectangular irradiance distribution were produced based on the arrangement of irradiance arrays and LED lens; a large-scale uniform colorful illumination could then be achieved through the irradiance distribution of the array [7].

Based on the aforementioned studies, structural design frequently uses light pipes, diffusers, micro-lens arrays, and other optical elements to assist in color mixing and achieve the best color-mixing effect. These optical elements may result in optical output power loss although they may facilitate color mixing. Therefore, the current study aims to propose a method in which the reflection principle of a reflector will be adopted to extend the irradiation distance of an optical structure. The sag equation will be used to design the curve surface of reflective mirrors such that the structural design will not require extra optical elements and will not result in optical output power loss. Consequently, output illuminance and the color-mixing effect will be simultaneously optimized.

2. Establishing the experimental steps and the computation model

The optical computation flow is illustrated in Fig. 1. First, lamp construction is considered by establishing the flashlight and the light source, designing the first and second reflecting surfaces, and building the geometrical structure model. These steps will be completed using SolidWorks computer-aided design software (Dassault Systèmes, France). Second, the color-mixing position of RGB LED will be entered and its luminous ratio will be altered. Third, the general uniformity of illuminance and the average difference of uniformity will be calculated. If the general uniformity is higher than 0.5 and the average difference of uniformity is lower than 30%, then we will return to the first step to change the lamp structure. Fourth, the uniformity of illuminance and color uniformity will be optimized for 17 detection points. Fifth, the simulation results will be outputted, and the uniformity of illuminance of the lamp and the color distribution range after light mixing will be analyzed. Finally, the design of the flashlight will be completed. Different views of the flashlight structure are shown in Fig. 2. Figure 2(a) shows the side view along the x–z plane, whereas Fig. 2(b) shows the side view along the z–y plane. The top view of the flashlight structure is along the x–y plane, as shown in Fig. 2(c). In this figure, S_R1 and S_R2 are the first and second reflecting surfaces that correspond to the red LED, respectively; S_G1 and S_G2 is the first and second reflecting surfaces that correspond to the green LED, respectively; S_B1 and S_B2 is the first and second reflecting surfaces that correspond to the blue LED, respectively; L_R is the red LED; L_G is the green LED; L_B is the blue LED; D_S12 refers to the distance between the first and second reflecting surfaces; and D_LS2 denotes the distance between a LED and the second reflecting surface.

Fig. 1 Establishment of the lamp module and flowchart of optical calculation.

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Fig. 2 Flashlight structure: (a) side view of the x–z plane, (b) side view of the z–y plane, and (c) top view of the x–y plane.

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By magnifying the side view of the second reflecting surface, an included angle θ₁ is found to exist between the second reflecting surface and the x–y plane, as shown in the diagrams at the lower part of Figs. 2(a) and 2(b). The magnified top view is shown in the diagram at the lower part of Fig. 2(c), where R_S2 refers to the radius of the second reflecting surface of the red LED, D_OS2 denotes the distance between the second reflecting surface and the center of the flashlight, D_S2BR denotes the distance between the center of the second reflecting surface of the blue LED and that of the red LED, D_S2RG refers to the distance between the center of the second reflecting surface of the red LED and that of the green LED, and D_S2GB denotes the distance between the center of the second reflecting surface of the green LED and that of the blue LED. The light ray tracing of the entire system is shown in Fig. 3. The LED illuminates the first reflecting surface with a small divergence angle; this surface causes the LED light rays to become parallel and then reflects them onto the second reflecting surface. The best curve surface for the second reflecting surface is determined after the computation. This curve surface can expand the parallel light rays from the first reflecting surface to the working plane located 10 cm away from the flashlight. Finally, a series of analyses and discussion based on the working plane will be conducted.

Fig. 3 Light ray tracing diagram of RGB LED within the flashlight.

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3. Luminance and color uniformity

The uniformity of illuminance is also known as the uniformity of light distribution because it defines the uniformity of light distribution within the illumination zone. The uniformity of illuminance value serves as a reference when the uniformity degree of illuminance distribution within a specific area is being identified. A uniform distribution of light is essential because the eyes may feel tired when they view objects for a long time if light is unevenly distributed, which, in turn, can lead to low work efficiency. The uniformity of illuminance can be identified in two ways: general uniformity and the average difference of uniformity. These approaches are discussed in the following sections.

3.1 General uniformity

General uniformity is also known as distribution uniformity. It refers to the ratio between the minimum and average luminance of space with a value between 0 and 1 [8–10], with 1 is the ideal value.

Uniformity = \frac{E_{0}}{E_{a}}

where E₀ denotes minimum luminance in space, and E_a refers to average luminance in space.

3.2 Average difference of uniformity

The average difference of uniformity is defined as the ratio between luminance and the average luminance difference of all pixels; it is expressed in percentage. If the value approximates 0, then the uniformity of illuminance in space is high; if the value is high, then the uniformity of illuminance in space is low. Equation (2) can be used to compute the average difference of uniformity [11].

E = \frac{1}{N} \sum_{n = 1}^{N} | \frac{E_{n} - E_{a}}{E_{a}} | \times 100 %

where E_n refers to the illuminance value of the sampling points, N denotes the number of sampling points, and E_a is the average illuminance value on the working plane.

3.3 CIE 1976 color uniformity

Color uniformity is generally measured by the “non-uniformity” of color distribution, that is, color uniformity will be good if non-uniformity is low. Color uniformity is defined as the root-mean-square value of the difference between the sampling points u, v of CIE 1976 and the average u, v. Equation (3) is used to calculate color uniformity, and the quantity of sampling points is M. The configuration mode of the 37 sampling points in the paper published by Sun et al. is adopted in this study [5]. Figure 6(c) in [5] shows the position of the sampling points on the working plane. This previous study indicates that measurement for a circular grid configuration is easier than that for a square configuration. On the one hand, the points next to the center of the light figure significantly affect the field of view of the observer because the points next to the circular grid center are dense. On the other hand, the square grid configuration is incompatible with the circular illumination.

Δ u' v'_{r m s} = \sqrt{\frac{1}{M} \sum_{i}^{M} [{(u_{i}^{'} - u_{a v g}^{'})}^{2} + {(v_{i}^{'} - v_{a v g}^{'})}^{2}]}

where M is the number of sampling points of the illuminated surface, and uʹ and vʹ are the color coordinates in the CIE 1976 uniform color system. We use the coordinates of the average color of the measured points, namely, uʹ_avg and vʹ_avg, as reference color points.

4. Design concept of a reflective mirror

A reflective mirror is designed to improve the service efficiency of light and to alter its direction. The mirror leads the light ray to the desired direction and enables the light to concentrate on the working plane. Consequently, the uniformity of illuminance and color uniformity within a certain scope reach their optimal levels.

4.1 Mathematical formula and relative position of the first reflective mirror

The design of a 3D reflector is complicated. Thus, we design a reflective mirror by rotating the conic section to achieve symmetry. Based on the conic section curves in [12], each curve has a focus point F [12], and the sag equation can be used to set different conic surfaces. This equation is shown as Eq. (4). For the conic constants of the conic sections, interested readers can refer to the table of characteristics of the five main types of conic surfaces in [12.

z = s a g (r) = \frac{r^{2} / R}{1 + \sqrt{1 - (1 + k) r^{2} / R^{2}}}

Where

r^{2} = x^{2} + y^{2}

, R denotes the radius of curvature, z refers to the Z-axis height difference, and k is the conic constant.

If the reflective mirror has a parabolic shape and the light source is placed at the focus point, then the light reflected by the parabolic surface will form parallel rays. The elliptical mirror will focus the light. The light ray coming from the focus point of a hyperbola will diverge after being reflected by the hyperbola. However, the spherical reflector may reflect the light back to the focus point. Based on the aforementioned results, parallel light can be formed if the light source is placed at the focus point of a parabolic reflector. Therefore, the first reflective mirror uses the parallel light function of the parabolic mirror. To enable the parabolic mirror to reflect all the LED lights onto the second reflective mirror, we first define the bottom center of the flashlight as the origin point (0, 0), rotate the red LED toward the Z₁-axis by 30°, and then rotate the red LED and the first reflective mirror toward the coordinate (–40, 60) by 19° to allow the light to be illuminated at the second reflecting surface, as shown in Fig. 4(a). In the entire frame chart of the flashlight, the red LED and its first reflective mirror S_R1 are placed at (–40, 60) of the Z–X coordinate system of the flashlight. The curvature radius is 40 mm, the opening size is 16 mm, and the reflectivity of aluminum is 0.9071, as shown in Fig. 4(b). We then add the green LED and its first reflective mirror S_G1 at the same coordinate after rotating the Z-axis by 120°. The curvature radius is 40 mm, the opening size is 16 mm, and the reflectivity of aluminum is 0.9136, as shown in Fig. 4(c). Finally, we add the blue LED and its first reflective mirror S_B1 at the same coordinate after rotating the Z-axis by 240°. The curvature radius is 40 mm, the opening size is 16 mm, and the reflectivity of aluminum is 0.9204, as shown in Fig. 4(d). We obtain Eq. (5) when k = −1 and R = 40 mm. We then convert this equation into the equation of the X–Y axes using the coordinate transform formula, as shown in Eq. (6).

Fig. 4 (a) Diagram of the 30° rotation of LED, (b) relative position of the reflective mirror S₁₁ of the red LED, (c) relative position of the reflective mirror S₁₂ of the green LED, (d) relative position of the reflective mirror S₁₃ of the blue LED.

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z 1' = s a g (x 1') = \frac{x 1'^{2}}{80}

z = - (263 + 4 x) \pm \sqrt{5 x^{2} + 2650 x + 1.246 \times 10^{5}}

4.2 Mathematical formula and relative position of the second reflective mirror

First, the second reflective mirror is assumed as a plane mirror. We determine its position relative to the first reflecting surface. The opening diameter of the first aspheric mirror is significantly related to that of the second aspheric mirror. If the diameter of the first aspheric mirror is larger than that of the second aspheric mirror, then we will assume that the opening diameter of the second reflective mirror is 18 mm. To prevent the reflective mirrors from affecting the light route of the other, we solve the origin point of the second reflective mirror through trigonometric identity. Finally, we place the second reflective mirror S₂ at (−15.3, 0) of the Z–X coordinate system, and the distance between reflective mirrors S₂ and S₁ is 63.5 mm, as shown in Fig. 5(a). In addition, based on the law of reflection, if reflective mirror S₂ is parallel with the X–Y axes, then the light will illuminate coordinate (0, 44.4) after being reflected by S₂, as shown in Fig. 5(b). Based on the law of reflection and trigonometric identity, the light can illuminate the (0, 330) plane if we rotate the Z₂–X₂ axes by 8.17°, as shown in Fig. 5(c). Therefore, we analyze the general uniformity and the average difference of uniformity at 0.1 mm interval within the curvature radius range of 80–125 mm using a hyperbolic mirror (k = −1.5), a parabolic mirror (k = −1), a transversely elliptical mirror (k = −0.5), a spherical mirror (k = 0), or a vertically elliptical mirror (k = 0.5) as the second reflective mirror. The results are shown in Figs. 6(a) and 6(b), where the vertical axis is the conic constant, whereas the horizontal axis is the curvature of the second reflective mirror. The color bar shows the values of general uniformity and the average difference of uniformity. According to the results of general uniformity, general uniformity is the highest and the average difference of uniformity is the lowest when the spherical mirror with a curvature of 105 mm⁻¹ and k = 0 is used as the second reflective mirror. Based on the results, the spherical mirror with a curvature of 105 mm⁻¹ is the best choice for the second reflective mirror, and the expression is shown as Eq. (7):

Fig. 5 (a) Relative positions of reflective mirrors S₂ and S₁, (b) diagram of LED light being reflected to coordinate (0, 44.4), (c) diagram of LED light being reflected to coordinate (0, 330).

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Fig. 6 (a) Relative position of reflecting surfaces S₂ and S₁, (b) diagram of LED light being reflected to coordinate (0, 44.4).

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z 2' = s a g (x 2') = \frac{x 2'^{2} / 105}{1 + \sqrt{1 - x 2'^{2} / 105^{2}}}

5. Illuminance on the working plane and the color-mixing results

To achieve the best design outcome, we set the fixed variables as k = 0 and R = 105 mm; the optimized variables as k₁, h₂, k₂, θ₁, and θ₂; and optimize the illuminance of 37 points and the average value of the working plane. The results are shown in Table 1. As shown in this table, the optimized variable that significantly changes is the height of the first reflecting surface. By contrast, the other variables insignificantly change. According to the results, general uniformity increases, whereas the average difference of uniformity declines, and the displacement of the chromaticity coordinate is less than 0.0044. Finally, optimizing color uniformity is unnecessary because such uniformity has already been optimized after the uniformity of illuminance is optimized. In analyzing color mixing, we set the luminous flux ratio among RGB LEDs and solve the color mixing results under 8 different correlated color temperatures of a blackbody radiator. The correlated color temperatures of white light are 2700, 3000, 3500, 4000, 4500, 5000, 5700, and 6500 K. After the computation results are combined with the chromaticity specification of solid-state lighting products, as shown in Fig. 7 [13], the color-mixing distribution of the flashlight under different correlated color temperatures agrees well with the scope of each color temperature. To determine the proper mixture source spectra, we referred to specifications on lamp illumination of the ANSI [14]. As shown in Fig. 7, this process can start from the 6- or 7-step MacAdam ellipses. A “step” indicates standard deviation [15]. During the first step, 68.2% of observers can repeatedly distinguish color differences, whereas during the second step, the proportion is 95%. ANSI suggested that manufacturers can produce fluorescent lamps within 4-step MacAdam ellipses. However, the scope of 4-step MacAdam ellipses was minimal, thus the Department of Energy expanded it to 7-step MacAdam ellipses [16]. In 2008, however, as the popularity of LED increased, 7-step MacAdam ellipses was no longer acceptable to manufacturers. Consequently, ANSI published a rectangular scope at various specific color temperatures of 2700, 3000, 4000, 4500, 5000, 5700, and 6500 K. Table 2 shows the computation results of the proposed optical structure under the 8 correlated color temperatures. As shown in this table, the average illuminance can be maintained at approximately 4200–4300 lux under the white light of different color temperatures. The average illuminance can also be maintained at the same level when color cast illumination is used, and the light intensity of the flashlight will not weaken when we adjust the RGB ratio. Subsequently, general uniformity and standard uniformity are compared with the results of previous studies, in which we have made a light steel frame fluorescent lamp, a LED lamp, and a new type of light steel frame LED lamp [8]. The general uniformity of the light steel frame fluorescent lamp is approximately 0.6–0.7. The value obtained by the present study is 0.68, which is slightly lower than that of the light steel frame florescent lamp. The optical design in the present study is used for local illumination within a short distance; hence, reaching the same level of uniformity as that of the light steel frame fluorescent lamp is unnecessary. Regarding the average difference of uniformity, several scholars have calculated such difference for indoor illumination lamps [11]. Their computation results for a lamp made of fluorescent tube or LED and reflective mirrors are 41%, 30%, and 10%. The best result is 10%; however, this result is attributed to the use of a diffuser. A diffuser is not used in our design and our average difference of uniformity is approximately 25%. Although this value is not comparable with that of the design with a diffuser, it is better than those of other designs without a diffuser.

Table 1. Positions of the optical system and optical values before and after optimization.

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Fig. 7 Graphical representation of the chromaticity specification of solid-state lighting products on the CIE 1931 chromaticity diagram with combinatorial numbers of LED color mixture types.

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Table 2. Optical computation results of the RGB LED flashlight under eight correlated color temperatures

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6. Conclusion

Based on the previously discussed optical design, our LED flashlight does not require a diffuser or any diffusing component to assist in light or color mixing. Each monochrome LED is provided with two reflective mirrors. The first reflecting surface can produce parallel light rays, whereas the second reflecting surface can expand the light beams. The RGB light then passes through the light-mixing distance, and finally, 8 different correlated color temperatures are successfully obtained. The outcome of the light-mixing design is as follows: general uniformity is approximately 0.68, the average difference of uniformity is approximately 25%, and color uniformity is roughly 0.0042. Such flashlight can be used for white light illumination of different color temperatures and for special color cast illumination, such as oral cavity illumination, museum illumination, and diving illumination.

Acknowledgment

This research was supported by the Ministry of Science and Technology, The Republic of China, under the Grants MOST 103-2622-E-194-010-CC3 and 104-2221-E-194-054.

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Position	Before optimization	After optimization
h₁	60 mm	82 mm
W₁	40.0 mm	41.8 mm
θ₁	19.0°	17.3°
h₂	0 mm	0 mm
W₂	13.5 mm	11.4 mm
θ₂	8.17°	6.44°
Uniformity	0.568	0.595
Average difference of uniformity	29.87%	27.45%
Chromaticity coordinates	(0.3445,0.3465)	(0.3401,0.3432)

	2700K	3000K	3500K	4000K	4500K	5000K	5700K	6500K
Illumination distribution of working face
Color mixing of working face
Chromaticity diagram	(0.4559, 0.4157)	(0.4381, 0.4081)	(0.4010, 0.3969)	(0.3752, 0,3872)	(0.3561, 0.3618)	(0.3401, 0.3432)	(0.3178, 0.3449)	(0.3055, 0.3278)
Minimum illumination values	2922 lux	2911 lux	2913 lux	2917 lux	2929 lux	2918 lux	2919 lux	2996 lux
Average illumination values	4248 lux	4257 lux	4279 lux	4315 lux	4320 lux	4324 lux	4337 lux	4349 lux
Total flux	10.62 lm	10.64 lm	10.69 lm	10.78 lm	10.80 lm	10.81 lm	10.84 lm	10.87 lm
Uniformity	0.685	0.684	0.681	0.676	0.678	0.675	0.673	0.689
Average difference of uniformity	25.53%	25.79%	25.91%	26.38%	26.97%	27.45%	28.17%	28.46%
Color Uniformity $Δ u^{'} {v^{'}}_{r m s}$	0.00427	0.00431	0.00428	0.00433	0.00436	0.00429	0.00434	0.00431

Position	Before optimization	After optimization
h₁	60 mm	82 mm
W₁	40.0 mm	41.8 mm
θ₁	19.0°	17.3°
h₂	0 mm	0 mm
W₂	13.5 mm	11.4 mm
θ₂	8.17°	6.44°
Uniformity	0.568	0.595
Average difference of uniformity	29.87%	27.45%
Chromaticity coordinates	(0.3445,0.3465)	(0.3401,0.3432)

	2700K	3000K	3500K	4000K	4500K	5000K	5700K	6500K
Illumination distribution of working face
Color mixing of working face
Chromaticity diagram	(0.4559, 0.4157)	(0.4381, 0.4081)	(0.4010, 0.3969)	(0.3752, 0,3872)	(0.3561, 0.3618)	(0.3401, 0.3432)	(0.3178, 0.3449)	(0.3055, 0.3278)
Minimum illumination values	2922 lux	2911 lux	2913 lux	2917 lux	2929 lux	2918 lux	2919 lux	2996 lux
Average illumination values	4248 lux	4257 lux	4279 lux	4315 lux	4320 lux	4324 lux	4337 lux	4349 lux
Total flux	10.62 lm	10.64 lm	10.69 lm	10.78 lm	10.80 lm	10.81 lm	10.84 lm	10.87 lm
Uniformity	0.685	0.684	0.681	0.676	0.678	0.675	0.673	0.689
Average difference of uniformity	25.53%	25.79%	25.91%	26.38%	26.97%	27.45%	28.17%	28.46%
Color Uniformity $Δ u^{'} {v^{'}}_{r m s}$	0.00427	0.00431	0.00428	0.00433	0.00436	0.00429	0.00434	0.00431

All-reflective RGB LED flashlight design for effective color mixing

Abstract

1. Introduction

2. Establishing the experimental steps and the computation model

3. Luminance and color uniformity

3.1 General uniformity

3.2 Average difference of uniformity

3.3 CIE 1976 color uniformity

4. Design concept of a reflective mirror

4.1 Mathematical formula and relative position of the first reflective mirror

4.2 Mathematical formula and relative position of the second reflective mirror

5. Illuminance on the working plane and the color-mixing results

6. Conclusion

Acknowledgment

References and links

Cited By

Figures (7)

Tables (2)

Equations (7)

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