Introduction

Nowadays, due to excellent characteristics, such as long lifetime, energy saving, environment friendly, and high color rendering, light-emitting diodes (LEDs) have been successfully used in the area of indoor/outdoor lighting, vehicle headlights, backlight of liquid crystal display, landscape lighting, search light, and so forth [1,2,3,4]. After years of development, a blue LED chip coated with YAG:Ce phosphor material has became one of the most popular commercial white LED generation approaches [5,6,7,8,9,10,11,12]. The basic principle of this type of white LED is that a portion of blue light emitted from LED chip is absorbed by YAG:Ce phosphor material, which generates yellow light, then the generated yellow light mixes with the remaining blue light, forming the white light. Generally, without exerting special treatment, the angular color uniformity (ACU) of the initially designed phosphor-converted LED (pc-LED) is poor. This is due to the fact that blue light rays with different emitting angles undergo different optical paths when passing through the YAG:Ce coating, the longer the optical path is, the more the blue light is depleted, finally, the ratios of blue to yellow light are different when viewing the LED light source from different viewing angles [13,14,15,16,17,18,19]. A number of methods to enhance the ACU of the pc-LED, such as using freeform optics or elaborately devising the structure of phosphor coating, had been intensively studied [13,14,15,16,17,18,19,20,21,22,23,24]. However, these works did not discuss in more details about the illumination uniformity, which is another important issue in general illumination applications. In other words, additional secondary optical elements are needed to redistribute the light energy emitted from LED to further improve the illumination performance of the lighting system.

Compared with the methods of devising the phosphor coating structure, freeform optics approaches seem to be more favorable to improve the optical performance of pc-LED based lighting system [20,21,22,23,24], since it can be used not only to improve the ACU of the system, but also to achieve a uniform illumination on the target plane [25,26,27,28,29]. Li et al. [26] developed a modified total internal reflection (TIR) lens to address the ACU issue of pc-LED and commendable color and illumination uniformity was obtained. Lai et al. [27] designed a freeform lens for small-angle lighting and uniform distribution of correlated color temperature (CCT) and illumination was achieved. Prins et al. [28, 29] proposed an inverse method for TIR collimators to achieve a uniform color spot light with prescribed intensity distribution in the far field. However, these works were mainly devised for lighting units with small cut-off angles. Different from above-mentioned studies, a novel design scheme aims to simultaneously address both the illumination and color uniformity of pc-LED based illumination system is developed. The key point of this work is combining a primary packaging freeform lens and an aspheric reflector in illumination system design. These two components can not only modulate the light energy emitted from LED to generate a light spot with uniform illuminance distribution, but also mix the blue and yellow light adequately to make the light spot has a uniform color distribution. In addition, this design scheme can achieve luminaire with a specified cut-off angle within a wide cut-off angle range.

Optical model of pc-LED

As shown in Fig. 1a, a typical pc-LED consists of a substrate, a packaging lens, and an LED chip coated with phosphor material. The LED light source is immersed in the packaging lens [13,14,15, 22,23,24]. A portion of blue light emitted from the LED chip is absorbed by the phosphor material and yellow light is yielded by photoluminescence effect, then the yellow light mixes with the remaining blue light to produce white light. Fig. 1b shows the excitation and emission spectra of the phosphor material, and spectrum of the blue LED that used to construct the optical model of the pc-LED in this work [30]. It is assumed that the LED chip has a dimension of 0.7 mm × 0.7 mm and its average refractive index is 2.5, the light rays were emitted from the top surface of the LED chip; the phosphor layer was coated on the LED chip and its dimension is 1.0 mm × 1.0 mm, the phosphor particles were approximated as spheres with diameters vary in the range of 1 to 15um; the phosphor powder was mixed with a transparent material with a refractive index of 1.5, the material of the packaging lens is polymethyl methacrylate (PMMA) with a refractive index of 1.49. More details about the construction of pc-LED can be found in [22, 31,32,33].

Fig. 1
figure 1

a pc-LED architecture and b excitation and emission spectra of phosphor material, and spectrum of the blue LED

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Figure 2a shows the simulated real color light pattern of the pc-LED with an average CCT of about 5000 K and Fig. 2b shows the corresponding illuminance and CCT distributions across centre section (u = 0) of the light pattern. One can find out easily that the CCT in the centre region is larger than that in the edge and more than 4000 K CCT difference can be noticed from centre to edge. Furthermore, a yellow light ring can be seen around the pattern, the simulation result is similar to the results shown in the previous works [14, 15, 21,22,23,24, 31, 32].

Fig. 2
figure 2

a Real color light pattern of the modeled LED light source with an average CCT of about 5000 K, and b shows the corresponding illuminance and CCT distributions across centre section (u = 0)

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Figure 3 shows the light intensity distribution curve of the modeled pc-LED and it illustrates that the light source is close to an ideal Lambertian emitter. For the purpose of obtaining precise simulation results in the following, the light intensity distribution of the pc-LED was represented by a polynomial function [34, 35], as expressed in eq. (1):

$$ {I}_s\left(\varphi \right)={I}_0\left({a}_0+{a}_1{\varphi}^2+{a}_2{\varphi}^4+{a}_3{\varphi}^6+{a}_4{\varphi}^8\right) $$
(1)

where φ represents the angle that the emitted light ray makes with the optical axis (the line vertical to the top surface of LED chip). I0 is the luminous intensity at φ = 0°. ai (i = 0, 1, 2, 3, 4) represent the coefficients of the polynomial function and the numeric values are a0 = 0.9707, a1 = − 0.6318, a2 = 0.0536, a3 = 0.05509, a4 = − 0.01422, respectively.

Fig. 3
figure 3

Simulated and fitted light intensity distribution curves of the modeled LED source

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Design method

Figure 4 shows the geometrical structure of the developed LED luminaire. The primary packaging lens, into which a pc-LED light source is immersed, is a freeform lens that made by PMMA optical material [22, 23]. It is assumed that the LED source is placed at the origin of the coordinates (represent by (u, v, z) and v is in the outward direction perpendicular to the plane). The aspheric reflector is made by metallic material and the inside surface of it is coated with reflective material with a reflectivity of 95%. a and b represent the semi-aperture size and the depth of the reflector, respectively. φcut denotes the cut-off angle of the luminaire, it is generally given by designers according to illumination requirements. (ul0, zl0) and (ur0, zr0) represent the initial points of the freeform lens and the aspheric reflector, respectively. (ul0, zl0) divides the surface of the lens into two parts: surface 1 (S1) and surface 2 (S2). It is worth noting that S1 and S2 are connected smoothly at the connecting point (ul0, zl0). The angle φc divides the light energy emitted from the LED into two parts: light energy in the angle range of 0° to φc and in the range of φc to π/2. The S1 of the freeform lens was designed to deflect the light rays in the range of 0° to φc toward the range of 0° to φcut to generate a light pattern with uniform light distribution on the target surface. In order to generate a light pattern with uniform color distribution on the target surface, it is important to mix the yellow and blue light adequately. Thus, the S2 of the lens was designed to deflect the light rays in the angle range of φc to π/2 toward the aspheric reflector, and then the reflector reflected these rays to the range of φcut to 0° to generate a light pattern as the previous one. The aforementioned contents imply that the light distribution on the target region was formed by two uniformly distributed light patterns, which can be mathematically expressed as:

$$ 2\pi {\int}_0^{\varphi_c}{I}_s\left(\varphi \right)\sin \varphi d\varphi =2\pi {\int}_0^{\varphi_{cut}}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi $$
(2)
$$ 2\pi \rho {\int}_{\varphi_c}^{\pi /2}{I}_s\left(\varphi \right)\sin \varphi d\varphi =2\pi {\int}_0^{\varphi_{cut}}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi $$
(3)

where Is(φ) and Ils(φ) denote the light intensity distributions of the LED light source and the designed luminaire, respectively. ρ denotes the reflectivity of the reflective material. For a uniform far-field illumination, the light intensity distribution of the lighting unit can be expressed as [35, 36]:

$$ {I}_{ls}\left(\varphi \right)=\frac{I_{l0}}{\cos^3\left(\varphi \right)} $$
(4)

where Il0 denotes the luminous intensity of the lighting unit at φ = 0°.

Fig. 4
figure 4

The structure of the developed LED luminaire. a and b represent the semi-aperture size and the depth of the reflector, respectively; φcut represents the cut-off angle of the luminaire, (ul0, zl0) and (ur0, zr0) represent the initial points of the freeform lens and the aspheric reflector, respectively; (ul0, zl0) divides the surface of the lens into two parts: surface 1 (S1) and surface 2 (S2), the two surfaces are connected smoothly at the connecting point; φ, with subscript, or both superscript, denotes the angle that light ray makes with optical axis, Δφ and Δβ denote the angle increment of emitting and refractive light ray, respectively

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In order to construct the freeform lens and the aspheric reflector, the relations between the incident rays and the outgoing rays at the two surfaces should be obtained. For the light ray with the emitting angle φi within the angle range of 0° to φc, since it only passes through the freeform lens, the relation between the emitting ray angle φi and the corresponding outgoing ray angle φ’i at the lens surface can be obtained by eq. (5). For the light ray with emitting angle φri within the angle range of φc to π/2, since it is first refracted by the freeform lens and subsequently reflected by the aspheric reflector, the relation between the emitting ray angle φri and the corresponding reflected outgoing ray angle φ’ri can be obtained by eq. (6).

$$ \frac{2\pi {\int}_0^{\varphi_{cut}}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi}{2\pi {\int}_0^{\varphi_c}{I}_s\left(\varphi \right)\sin \varphi d\varphi}=\frac{2\pi {\int}_0^{\varphi {\hbox{'}}_i}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi}{2\pi {\int}_0^{\varphi_i}{I}_s\left(\varphi \right)\sin \varphi d\varphi} $$
(5)
$$ \frac{2\pi {\int}_0^{\varphi_{cut}}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi}{2\pi \rho {\int}_{\varphi_c}^{\pi /2}{I}_s\left(\varphi \right)\sin \varphi d\varphi}=\frac{2\pi {\int}_0^{\varphi {\hbox{'}}_{ri}}{I}_{ls}\left(\varphi \right)\sin \varphi d\varphi}{2\pi \rho {\int}_{\varphi_{ri}}^{\pi /2}{I}_s\left(\varphi \right)\sin \varphi d\varphi} $$
(6)

Up to present, the only unknown quantity in construction of the two surfaces is φti, as shown in Fig. 4, which represents the refraction angle of outgoing light ray corresponding to the incident light ray with emitting angle larger than φc. Keeping in mind that the emitting angle φri of the light ray is increased step by step by a small angle increment Δφ in constructing the freeform lens, thus, it is instinctively assumed that the corresponding outgoing ray angle φti is also increased step by step by a small increment angle Δβ = (π/2-φc)/M, where M is a positive integer which denotes the number of partitions. As φti = φc + iΔβ is known, together with the emitting light ray angle φi and φri, refracted outgoing ray angle φ’i, reflected outgoing ray angle φ’ri, and the given initial points of the freeform lens and aspheric reflector, the lens and reflector can be constructed easily [34, 37].

Simulation and discussion

Influence of φc on color uniformity

Figure 2 shows that both the illuminance and the CCT distributions decrease nonlinearly from centre to edge on the receiver as a bare LED light source is used in the illumination system. To design a system with acceptable optical performance using this white LED source, both the illumination and color uniformity should be considered in the design process. The uniformity of lighting on the target surface is ensured by eqs. (5) and (6), the dependence of color uniformity on φc will be discussed in the following. In this section, the cut-off angle φcut of the luminaire is set to 50°, and φc is changed from 40° to 50°. As the initial points ul0 of the lens and ur0 of the reflector are given, the illumination system can be constructed. In order to facilitate comparison, ul0 was selected to make the centre height of the lens to be 6 mm for each φc, and ur0 was set to 20 mm. Figure 5a and b show the profiles of the designed luminaires with different energy partition angle φc and the three dimension model of the luminaire with φc = 45°, respectively. One can note from the figure that the shapes of the aspheric reflectors corresponding to different energy partition angle φc are almost identical and the small difference in the geometrical sizes of the reflectors is caused by the difference in selection of ul0 of the lens. One can also notice from the figure that as the centre height of the lens remains constant, the total geometrical size of the freeform lens decreases with the increase of φc.

Fig. 5
figure 5

a Profiles of the designed luminaires with different φc and b the three dimension model of the luminaire with φc = 45 °

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Figure 6 shows the simulation results of the designed luminaires with different energy partition angle φc. It is noted that the lines shown in Fig. 6a and b are sampled values across the centre (u = 0) of the target surface, and more detailed simulation results are shown in Table 1. It can be found from Fig. 6a that for a large energy partition angle φc, the CCT in the centre region of the target surface is larger than that in the edge; on the contrary, the CCT in the centre is smaller than that in the edge for a small φc. This result can be interpreted with the help of Fig. 2, one can note from the figure that the ratio of blue to yellow light in the centre is larger than that in the edge, since the light energy in the range of 0° to φc and φc to π/2 separately generate a uniform light pattern with same geometric size on the target surface, as φc increases, the quantity of yellow light reflected to the centre decreases, the CCT in the centre increases; conversely, the CCT in the centre decreases. As shown in Table 1, when φc = 45°, the variation of CCT across the centre of the target surface is smaller than 300 K [8, 38]. Figure 6b shows illuminance distribution corresponding to different energy partition angle φc, one can note that uniform illumination was obtained for each case; however, a very small difference exists in the extension of the illuminance distribution among these three cases, the reason is that the geometrical size ratios of the lens to the reflector are different for the three energy partition angle φc, as shown in Fig. 5a. Figure 6c shows the simulated real color light pattern of the luminaire with φc = 45 °, one can note that a uniform illumination, along with a uniform color distribution, was obtained.

Fig. 6
figure 6

Simulation results of the luminaires with different energy partition angle φc (a) CCT distribution, (b) illuminance distributions, and (c) simulated real color pattern of the luminaires with φc = 45 °

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Table 1 Simulation results of the luminaires with different energy partition angle φc and single lens
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Using the pc-LED source, an illumination system with a single freeform lens was also designed for comparison, as shown in Fig. 7. Figure 7a shows the profile of the single freeform lens, and the corresponding simulation results are shown in Fig. 7b and c. It can be found from Fig. 7b that although an uniform illumination is obtained on the target surface, the color distribution is very terrible, the CCT in the centre is higher than that in the edge and about 5600 K CCT variation can be noted from the centre to edge. Such big color variation is unacceptable in most of applications since uncomfortable visual effect will disturb human beings, as shown in Fig. 7c. It seems that the novel developed system has an additional reflector compared with single freeform lens based system, in fact, in most of applications, a reflector is an indispensable component of the illumination system, since it can be used not only to adjust light beams, but also to eliminate undesired stray light and/or to protect the light source. As shown in Fig. 8, without reflector, undesired stray light was generated by the single freeform lens, in contrast, the stray light can be blocked by the aspheric reflector of the novel system.

Fig. 7
figure 7

Simulation results of the luminaire with single freeform lens (a) profile of the lens, (b) illuminance and CCT distributions, (c) real color light pattern

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Fig. 8
figure 8

Ray tracing chart of (a) single freeform lens and (b) developed design scheme. Undesired stray light is unavoidable by only a single freeform lens, but it can be eliminated by the aspheric reflector of the novel system

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Design with different cut-off angle φcut

In different illumination applications, the luminaires with different cut-off angles are likely to be required, several examples within the cut-off angle range of 40° to 60° were designed. Figures. 9, 10, 11, and 12 show the simulation results of the luminaires with the cut-off angle of 40°, 45°, 55°, and 60°, respectively. Detailed simulation results were concluded in Table 2. After checking the simulation results, one can note that the energy partition angle φc decreases with the increase of φcut, one can also found that for each case, the illumination uniformity is better than 80%, the CCT variation is less than 300 K. These indices are within acceptable levels [8, 34, 38]. It is noted that the uniformity of illumination in this work is defined as the ratio of minimum illuminance to the average illuminance on the target surface. Figure 13 shows the light distribution curves of the designed luminaires with different cut-off angles, it can be seen that batwing-like distribution curves were obtained for all cases.

Fig. 9
figure 9

Simulation results of the luminaire with the cut-off angle of 40° (a) illuminance and CCT distributions, (b) real color light pattern

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Fig. 10
figure 10

Simulation results of the luminaire with the cut-off angle of 45° (a) illuminance and CCT distributions, (b) real color light pattern

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Fig. 11
figure 11

Simulation results of the luminaire with the cut-off angle of 55° (a) illuminance and CCT distributions, (b) real color light pattern

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Fig. 12
figure 12

Simulation results of the luminaire with the cut-off angle of 60° (a) illuminance and CCT distributions, (c) real color light pattern

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Table 2 Simulation results of the designed luminaires with different cut-off angle φcut
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Fig. 13
figure 13

Light intensity distribution of the designed luminaires

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Conclusion

In this study, a simple design scheme which can address both the problem of illumination and color uniformity of pc-LED based illumination system has been discussed. By employing only two ordinary optical components, i.e., a primary packaging lens and a reflector, the scheme can improve both the illumination and color performance of a pc-LED based lighting unit. Simulation results show that as proper energy partition angle φc was selected for a pre-specified φcut, better than 80% illumination uniformity, smaller than 300 K CCT variation can be obtained. Since the ACU of a pc-LED source mainly depends on the thickness variation of the phosphor layer, the design scheme showed in this study can increase the tolerances in coating the phosphor layer and reduce the cost of the illumination system to some extent. It is a useful alterative design method to improve the optical performance of pc-LED based lighting system in lighting industry. However, this study only focuses on the issue of illumination and color uniformity for a point-like pc-LED based illumination system, further extension to extended pc-LED based illumination system is required in the future work and this study is an important building block for more in-depth research.