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Design of optical system for collimating the light of an LED uniformly Chen Chen and Xiaohui Zhang* Ordnance Engineering Department, Naval University of Engineering, Wuhan, Hubei 430033, China *Corresponding author: [email protected] Received December 19, 2013; revised March 17, 2014; accepted March 31, 2014; posted April 2, 2014 (Doc. ID 203457); published April 30, 2014 A type of optical system consisting of one total internal reflection (TIR) lens and two reflectors is designed for collimating the light of an LED to a uniform pattern. Application of this kind of optical system includes underwater light communication and an underwater image system. The TIR lens collimates all the light of the LED to a nonuniform plane wavefront. The double-reflector system redistributes the plane wavefront uniformly and collimates again. Three optical systems that produce a different radius of the output light patterns are designed. The simulation result shows that the uniformity of the designed optical system is greater than 0.76, and the total output efficiency (TOE) is greater than 89%. At the same time, we conclude that the radius of the output reflector should not be smaller than that of the input reflector in order to keep high uniformity and TOE. One of the designed optical systems is fabricated by computer numeric control, and the experiment results satisfy that goal. © 2014 Optical Society of America OCIS codes: (080.2740) Geometric optical design; (080.4035) Mirror system design; (080.4295) Nonimaging optical systems; (220.2945) Illumination design. http://dx.doi.org/10.1364/JOSAA.31.001118
1. INTRODUCTION Compared with traditional light sources, the LED has many advantages such as high light efficiency, long life, small volume, high reliability, no warm-up period, and excellent color rendering [1]. Among those advantages, the small volume makes LED the first practical point light source. The illumination optical design has more flexibility, and it has received major progress due to this advantage. Parkyn [2] proposed an illumination lens design method, which establishes two grids on the source space and target space. Based on the grids, the illumination lens can be designed by the Snell’s law. Ries and Muschaweck [3] developed a method to tailor an optical surface based on energy conservation, differential geometry, and the Snell’s law. Using this method to design an optical system, several elliptical nonlinear partial differential equations must be solve. Parkyn and Pelka [4] applied a pseudo-rectangular spherical grid to establish correspondence between the source grid and target grid. With this correspondence, the coordinates of each point of lens can be calculated. With the development of freeform optical surface design, LED illumination optical systems became the hot spot of the research region. Wang et al. [5] proposed a design of discontinuous freeform lens, which is based on the variable separation mapping method. The discontinuous was introduced to control the normal deviation. Ding et al. [6] presented a freeform LED lens design method, which is based on tailored surface method and uses a source–target energy map to reduce the complexity of solving progress. Wang et al. [7] designed a freeform LED lens design based on the source–target energy map and analyzed how the installation errors affect the light pattern. Fournier et al. [7] derived a type of source–target map, which can generate a smooth freeform optical surface. Moiseev et al. [9] developed a method to design LED lens 1084-7529/14/051118-08$15.00/0
for elongated rectangular illumination. Wu et al. [10] constructed freeform optical surfaces using unit tangent vectors of feature data points as constrains. Wang et al. [11] presented a LED lens design, which combined a Fresnel lens and microlens array to generate uniform illumination. This method need not consider the light intensity of LED, but the manufacture of the microlens array is not easy. Moiseev and Doskolovich [12] first presented the analytical solution of total internal reflection (TIR) optical element to generate narrow-angle light distribution. The LED collimating lens design also has been developing constantly with the tool of freeform surface. Rubinstein and Wolansky [13] developed a method to design a freeform lens, which can redirect the light of a parallel beam to any prescribed light distribution. Chen and Lin [14] designed a freeform collimating lens and developed a modified freeform surface approximation method having more accuracy. Wang et al. [15] proposed a combination of a freeform lenscoupled parabolic reflector to solve the LED searchlight design problem. Wang et al. [16] designed two types of LED collimating lenses based on the theory of nonimaging optics and geometrical optics. Chen et al. [17] presented a freeform LED collimating lens method derived from basic geometric-optic, and this lens has better optical efficiency and smaller spot size compared with the lens in [13]. Aslanov et al. [18] proposed a compact optical system generating uniform irradiance distribution in a noncircular region. This optical system consists of a reflector and a microlens array. Using different shapes of the unit of a microlens array can generate any shape of illumination area. In an LED-based underwater optical detecting system, the nonuniformity of illumination causes the saturation on the center and dim on the edge. In order to overcome this © 2014 Optical Society of America
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shortcoming, the light pattern must be uniform and in a narrow view angle. In this paper, an optical system, which can collimate LED light uniformly, is proposed. This optical system consists of a TIR lens and a double-reflector system. The TIR lens collimates all the light of the LED parallel to the z axis but nonuniformly. Then Reflector 1 redirects the plane wavefront of the collimated light to uniform distribution, and Reflector 2 collimates the light again but uniformly. The simulation result shows this optical system can achieve illuminance uniformity more than 0.75, and the total output efficiency (TOE) of this system is 89.33%. We can control the radius of output light pattern through changing the radius of Reflector 2. In order to keep high uniformity and efficiency, the radius of Reflector 2 should not smaller than the radius of Reflector 1. This paper is organized as follows: Section 2 presents the design of the optical system. Section 3 presents the freeform construction process, which is based on the source–target energy map. Section 4 shows the simulation results and discusses some specific circumstances.
2. DESIGN OF THE OPTICAL SYSTEM Consider that the LED is a point source being collimated at a surface, which could be a refractive or a reflective one. In Fig. 1 light ray 1, 2, 3 refract at the optical surface and then parallel to the z axis. The light flux between light ray 1 and light ray 2 is equal to the light flux; therefore the light flux of the circle OR1 and the ring R1 R2 is equal. Because of the Lambertian light distribution of the LED, the angle θ1 is lager than θ2 . Hence the area of the circle is lager than the area of the ring R1 R2 , which causes unequal between the light distribution of the circle and the ring R1 R2 . So we can summarize that a single optical surface cannot collimate the light of the LED to a uniform light distribution. In order to achieve the uniform collimating of the LED, we propose the optical system demonstrated in Fig. 2. This optical system consists of three components: a TIR lens and two reflectors. The TIR lens collects and collimates all of the light of the LED to a plane but nonuniform wavefront. Reflector 1 redirects the nonuniform parallel light from the TIR lens to a uniform distribution on the reflector. Finally, Reflector 2 collimates the light to the uniform plane wavefront. Figure 3 shows the profile of the TIR lens, which has been used in [8]. This TIR lens has three freeform surfaces, which include two refractive and one reflective surface. Surface 1 directly refracts the light of the LED parallel to the z axis.
TIR
Reflector 1
Lens
Reflector 2 Fig. 2. Sketch map of the optical system.
But surface 1 cannot collimate all of the light of LED because of the TIR phenomenon, which can be overcome by Surface 2 and Surface 3. Surface 2 refracts the light to a bigger aspect angle, which guarantees that the TIR phenomenon will happen at Surface 3. Moreover the refractive light of Surface 2 can be regarded as the light emits from a point light source at f , which could be an arbitrary point at the negative x axis. The profile of the surface is a parabola whose focus point is point f . Thus the surface can reflect the light from Surface 2 parallel to the z axis. The points of surface can be calculated directly. Surface 1 and Surface 2 are freeform surfaces, which are constructed through the method presented in next section. As shown in Fig. 4, after being collimated by the TIR lens, the light of the LED is transformed to a plane wavefront, which is the input light beam of Reflector 1. The projection area of this wavefront at the plane z 0 is Ω. Then the light-intensity distribution of the input wavefront can be denoted as IΩ. The output wavefront of Reflector 2 is also a plane, and its projection area at the plane z 0 is T. Thus the light-intensity distribution of the output wavefront can be denoted as IT. The function IΩ is depended on the lightintensity distribution of the LED, and the function IT is a constant function, which is dependent on the whole light flux of the LED and the area of T. The calculation of IΩ will be shown in the next section, assuming all of the light of the input wavefront propagates to the output wavefront. For the conservation of energy, we can get the relationship between IΩ and IT as ZZ
R1
O
R2
1
ZZ IΩdΩ
L2
L1
ITdT:
(1)
z
2
θ1
3
3
θ2
1
2
x LED Fig. 1. Light of LED collimated by a single optical surface.
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f
LED
Fig. 3. Profile of the TIR lens.
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Reflector 2
The light-intensity distribution on a plane actually is illuminance, i.e., the total luminous flux incident on a surface per unit area. Denote E Ω and ET as the illuminance of the input and output wavefront. In the polar coordinate system, Eq. (1) changes into ZZ
E T r; θrdrdθ:
Z
ri
r i−1
0
Ω
(2)
Z E Ω r; θrdr dθ
2π
Z
Φ E T r; θrdr dθ : N r i−1
0
Output wave-front
In order to solve the shape data of the reflector numerically, the correspondence relationship between the points of the input and output wavefronts must be established. This correspondence relationship is based on the conservation of energy, i.e., Eq. (2). The input and output wavefronts are divided into same quantity of tiny patches, which have the same amount of light flux. Based on the edge principle [19], the light flux of the patches at the input wavefront will all propagate to the corresponding patches at the output wavefront if the edge rays of the patches at the input wavefront all propagate to the corresponding patches’ edge at the output wavefront. First we can obtain the relationship of the r in Eq. (2) of the input and output wavefront by doing integration along the r direction: 2π
wave-front
ZZ EΩ r; θrdrdθ
Z
Input
ri
T y
z A:
x
Z
ri r i−1
Z
θi
θi−1
Z E Ω r; θdθ rdr
ri r i−1
Z
Φ : E T r; θdθ rdr MN θi−1 θi
(4) In Eq. (4) M represents the number of patches along the θ direction in a ring. The edge points of these patches can be calculated through Eqs. (3) and (4). After that we can establish the one-to-one correspondence between the point of the input and output wavefronts.
3. FREEFORM SURFACE CONSTRUCTION A. Freeform Surface Construction of the TIR Lens Unlike Surface 3 in Fig. 3, a freeform surface could not be represented by an explicit function. To get a freeform surface, a large quantity of discrete points needs to be calculated one after another. For the TIR lens, we only need to calculate the profile and then revolve the profile around the z axis to get the shape of the lens. The reflectors are not symmetrical along the z axis; we need to calculate skeleton curves along the longitude direction then get the shape of the reflector by these curves. Figure 5 shows the process to construct freeform Surfaces 1 and 2 in Fig. 3, where the unit direction vector of the light ray emitting from the LED marked as I0 , I1 , I2 , and I3 hitting the freeform surface at P 0 , P 1 , P 2 , and P 3 , respectively, and O0 , O1 , O2 , and O3 are the unit direction vectors of the output light ray. In this figure, P 0 located at the z axis is the initial point,
Reflector 1 z
Fig. 4. Sketch map of the double reflector system.
which is set up according to the design require. The normal vector N0 at P 0 can be calculated by Snell’s law: 1 n2 − 2nO · I1∕2 N O − nI:
(5)
Then the tangent vector t0 at P 0 can be calculated by N0 . The next point of the lens profile P 1 is the point of intersection of the input light ray I1 and the tangent vector t0 . Assuming the angle between the light ray I1 and the z axis is θ1 , and the normal vector at P 0 is N0 N x ; N y . Then the coordinate of P 1 is
(3) In Eq. (3) Φ is the total luminous flux of the LED, and N is the number of patches along the r direction. Using Eq. (3) we get a series value of the radius. Then, doing the integration between two adjacent circles along the θ direction, we can get the relationship of θ between the input and output wavefronts:
=0
8
θth
7
Assuming the radius of the TIR lens is R then from Fig. 6, we can get the relationship of the radius of the TIR lens, and the radius of the input wavefront is
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O 0 O1 O 2
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1
O3 N3
N1 N 2
t2
t N0 t0 1
P3
P0 P1 P2
2
I 0 I1 I 2 I 3
LED Fig. 5. Construction process of the profile of the freeform surface of the TIR lens.
r R sin θ:
Using Eqs. (3), (4), (7), and (8), we can get the correspondence relationship between the points of input and output wavefronts. With this correspondence, the x; y coordinates of the two reflectors can be calculated. Figure 7 shows the construction process of the skeleton curves of the two reflectors. First choose the initial point of these reflectors, which are P 111 and P 211 . The superscripts 1 and 2 indicate which point belongs to which reflector, and the subscript 11 means this point is the first curve’s first point. The input vector at P 111 and output vector at P 211 are 0; 0; 1. The output vector at P 111 and input vector at P 211 are P211 − P111 ∕jP211 − P111 j. Then we can use Eq. (5) to calculate the normal vector at point P 111 and P 211 . Using these two normal vectors, we can calculate the tangent plane of these two points. The point of intersection between these two tangent planes and the next input light ray are the point P 112 and P 212 . Repeat this process until the last point of the curve is calculated. After the initial curves of the two reflectors are constructed, the rest of the skeleton curves of the reflector are calculated by the same procedure. First we calculate the normal vectors of the two initial curves. Then, using the tangent planes of the initial curves and the input light ray of next curve, we calculate the adjacent point of the next curve. The reflectors can be constructed by loft of the skeleton curves. According to the discussion in [8], the integrability of the source–target map affects the continuity of the freeform surface. In [5] the author believed that if there is a big discrepancy between the real normal vectors and the calculated Input wave-front of the reflectors
r
Fig. 7. Construct the skeleton curves of the reflectors.
(8)
rth
normal vectors, the irradiance on the target plane will differ from the prescribed irradiance. Figure 8 is the deviation between the real normal vectors and the calculated normal vectors of Reflector 1. The maximum value of the deviation is 5.6 × 10−3 , which is sufficiently small. Therefore we can construct the freeform surfaces of the reflector system without concern the discontinuity.
4. EXPERIMENT AND DISCUSSION After calculating the shape data of the TIR lens and the reflectors, the 3D model of the TIR lens and the reflectors were constructed in CAD software. Then the 3D models were imported to optical and illumination design software to validate the illumination pattern and the efficiency of the optical system. The CREE XR-E is used as the light source of this system. Figure 9(a) shows the 3D model of the TIR lens. We actually designed three reflector systems; each has a different radius of output reflector in order to generate different radius of light pattern. The radii of these reflectors are one times of TIR lens, -3
x 10 20
5
40 4
60 80
r
3
100 120
2 140 160
θth θr
3
1
180
1 θx
Fig. 6. Light propagates from the LED to the input wavefront.
200 50
100
150
200
250
300
0
Fig. 8. Deviation between the real normal vectors and the calculated normal vectors of Reflector 1.
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32.36mm
5mm
19.39mm
5mm 9.77mm
Reflector 1
Reflector 2
Fig. 9. Perspective view of the 3D model of the optical system. (a) TIR lens. (b) One time radius reflector.
1 0.9 0.8 0.7
Illuminance (lux)
two times of TIR lens, and 0.5 times of TIR lens. Figure 9 shows the 3D model of the one time radius reflector system. Figure 10(a) shows the light pattern after being collimated by the TIR lens, which is not as uniform as expected. Figure 10(b) shows the simulation result of the one time radius reflector system. The detecting surface is set to have the same x-y plane projective radius of Reflector 2 in Fig. 4. The uniformity of the light distribution at the detecting surface is defined as U E E min ∕Emax , in which E min and Emax represents the minimum and maximum illuminance of the detecting surface. From the relative illuminance distribution along the x and y axes of the detecting surface shown in Fig. 10(c), the uniformity across the x axis is U EX 0.78, and the uniformity across the y axis is U EY 0.82. As a comparison, the uniformity along the x and y axes of the TIR lens is U EX 0.0181 and U EY 0.0126. The TOE is defined as the ratio between the light flux of the output wavefront and the light flux of the LED. The ray trace result of Fig. 10(b) tells us that the TOE is 89%. Using the light efficiency of the TIR lens, we can calculate the light efficiency of the reflector system is 97%. Figure 11 shows the simulation result of two times radius reflector and one time radius reflector. The uniformity of two
0.6 0.5 0.4 0.3
X axis Y axis
0.2 0.1 0
-15
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-5
0
5
10
15
Distance to the center of the output plane (mm)
Fig. 10. (a) Light pattern collimated by TIR lens. (b) Light pattern after reshaping by the reflector system. (c) Illuminance along the x and y axes.
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0.1 0
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Fig. 11. Simulation result of different radius of reflector. (a) Two times radius reflector. (b) 0.5 times radius reflector.
0.6 0.5 0.4 0.3 0.2 X axis Y axis
0.1 0 -8
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Distance to the center of the output plane (mm)
Fig. 13. Simulation result of different radius of reflector using actual LED model. (a) Two times radius reflector. (b) One time radius reflector. (c) 0.5 times radius reflector.
source to simulate. The area of the light source in this Cree LED optical module is 1 mm × 1 mm. According to Fig. 8, the distance from the chip to Surface 2 of the TIR lens is less than five times the chip size. Therefore this actual Cree LED model could not be regarded as a point light source according to the far-field conditions of LEDs [20]. Figure 13 shows the 18000
0.5 times radius 1 times radius 2 times radius
0.5 times radius 1 times radius 2 times radius
14000
Intensity (Candle)
16000 14000
Intensity (Candle)
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Illuminance (lux)
times radius reflector in output plane along the x and y axes is U EX 0.76 and U EY 0.80. The TOE of two times reflector is 91%. From Fig. 11(b) we get the x and y axes uniformity of 0.5 times radius reflector being U EX 0.71 and U EY 0.63, the TOE of this reflector is 82%. The reason why the 0.5 times radius reflector has lower uniformity than the TOE is that the smaller radius of output reflector causes reduction of etendue of the system. In order to keep high uniformity and the TOE of the optical system, the radius of the output reflector should not smaller than the radius of input reflector. The luminous intensity distributions of these three optical systems are shown in Fig. 12. These three distribution curves almost overlap, which means the size of the reflector does not affect the luminous intensity distributions of the output beam. According to simulation results, the output beams of these optical systems have an FWHM of 2 × 2° . In order to investigate the optical performance of the design, we use the actual model of the Cree XR-E LED as the
0.9
FWHM = 2 × 2
8000 6000
12000 10000
FWHM = 2 × 4o
8000 6000 4000
4000
2000 2000 0 -30
0 -30 -20
-10
0
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Angle (degrees)
Fig. 12. Luminous intensity distributions of different optical systems.
-20
-10
0
10
20
30
Angle (degrees)
Fig. 14. Simulation results of the luminous intensity distributions of different optical systems using the actual LED model.
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Reflector2
TIR Lens
Reflector1
Fig. 15. Fabricated TIR lens and the two times radius reflectors.
simulations of these optical systems using the actual LED model. The uniformity of two times radius reflector in output plane along the x and y axes is U EX U EY 0.65. The TOE of two times reflector is 87%. The uniformity of the one time radius reflector in the output plane along x and y axes is U EX U EY 0.70. The TOE of two times reflector is 83%.
-30
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5. CONCLUSION
y(mm)
-10
0
10
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These uniformity and TOE are acceptable for the optical performance requirement. At the same time, the uniformity of the 0.5 times radius reflector in the output plane along the x and y axes is U EX U EY 0.59. The TOE of the 0.5 times radius reflector is 73%. The decline of the optical performance of the 0.5 times radius reflector is the worst among these reflectors. As per the discussion above, the reason for this is the decrease of the etendue. The actual model of the LED also affects the FWHM of the output beam. Figure 14 shows the luminous intensity distribution of these optical systems with the actual LED model. The maximum value of the 0.5 times radius reflector drops to 1.53 × 104 while the maximum values of the other two reflectors hardly have changed. Even though the FWHM of these three reflectors is nearly at the same value, which is 2 × 4°. As shown in Fig. 15, the TIR lens and the two times radius reflectors are fabricated by computer numerical control (CNC). The reflectors of this prototype are made of ABS, which was sputtered with an aluminum coating. The experimental result is shown in Fig. 16(a), and the pseudo color map of this result is shown in Fig. 16(b). Figure 16(b) shows clearly that there are some positioning errors during constructing the experimental setup. There are three main reasons: (1) the fabricated error of the CNC; (2) the coating error of the reflectors; (3) the installation error of the optical system. But, overall, we obtain a desirable design for the underwater optical-detecting system.
0
A single freeform surface lens cannot achieve uniform collimation of the LED light. In order to reach this goal, a double freeform reflector system is designed for this paper. This reflector system is designed by the source–target map method. First, the light of the LED is collimated by a TIR lens, and then the reflector system redistributes the light to a uniform light pattern and collimates the light again. The simulation result shows that the system can achieve 0.78 of uniformity and 89% TOE. We also investigate the optical performance of two other optical systems, which have different radii of the output reflector. The result shows that in order to keep high uniformity and light efficiency, the radius of the output reflector should not be smaller than the radius of the input reflector. Then, after simulation with an actual LED model, we discussed the impact of the size of the LED. Finally, we fabricated one of the designed optical systems. The experiment results show that the light pattern generated by this optical system satisfies our goal. The design of this paper can be applied in underwater photoelectric detecting systems such as underwater optical communications and underwater imaging.
0.5 0.4
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0 x (mm)
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Fig. 16. Results of the fabricated optical system. (a) Experimental result. (b) Pseudo color map of the experimental result.
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J. Opt. Soc. Am. A / Vol. 31, No. 5 / May 2014
C. Chen and X. Zhang
Design of optical system for collimating the light of an LED uniformly Chen Chen and Xiaohui Zhang* Ordnance Engineering Department, Naval University of Engineering, Wuhan, Hubei 430033, China *Corresponding author: [email protected] Received December 19, 2013; revised March 17, 2014; accepted March 31, 2014; posted April 2, 2014 (Doc. ID 203457); published April 30, 2014 A type of optical system consisting of one total internal reflection (TIR) lens and two reflectors is designed for collimating the light of an LED to a uniform pattern. Application of this kind of optical system includes underwater light communication and an underwater image system. The TIR lens collimates all the light of the LED to a nonuniform plane wavefront. The double-reflector system redistributes the plane wavefront uniformly and collimates again. Three optical systems that produce a different radius of the output light patterns are designed. The simulation result shows that the uniformity of the designed optical system is greater than 0.76, and the total output efficiency (TOE) is greater than 89%. At the same time, we conclude that the radius of the output reflector should not be smaller than that of the input reflector in order to keep high uniformity and TOE. One of the designed optical systems is fabricated by computer numeric control, and the experiment results satisfy that goal. © 2014 Optical Society of America OCIS codes: (080.2740) Geometric optical design; (080.4035) Mirror system design; (080.4295) Nonimaging optical systems; (220.2945) Illumination design. http://dx.doi.org/10.1364/JOSAA.31.001118
1. INTRODUCTION Compared with traditional light sources, the LED has many advantages such as high light efficiency, long life, small volume, high reliability, no warm-up period, and excellent color rendering [1]. Among those advantages, the small volume makes LED the first practical point light source. The illumination optical design has more flexibility, and it has received major progress due to this advantage. Parkyn [2] proposed an illumination lens design method, which establishes two grids on the source space and target space. Based on the grids, the illumination lens can be designed by the Snell’s law. Ries and Muschaweck [3] developed a method to tailor an optical surface based on energy conservation, differential geometry, and the Snell’s law. Using this method to design an optical system, several elliptical nonlinear partial differential equations must be solve. Parkyn and Pelka [4] applied a pseudo-rectangular spherical grid to establish correspondence between the source grid and target grid. With this correspondence, the coordinates of each point of lens can be calculated. With the development of freeform optical surface design, LED illumination optical systems became the hot spot of the research region. Wang et al. [5] proposed a design of discontinuous freeform lens, which is based on the variable separation mapping method. The discontinuous was introduced to control the normal deviation. Ding et al. [6] presented a freeform LED lens design method, which is based on tailored surface method and uses a source–target energy map to reduce the complexity of solving progress. Wang et al. [7] designed a freeform LED lens design based on the source–target energy map and analyzed how the installation errors affect the light pattern. Fournier et al. [7] derived a type of source–target map, which can generate a smooth freeform optical surface. Moiseev et al. [9] developed a method to design LED lens 1084-7529/14/051118-08$15.00/0
for elongated rectangular illumination. Wu et al. [10] constructed freeform optical surfaces using unit tangent vectors of feature data points as constrains. Wang et al. [11] presented a LED lens design, which combined a Fresnel lens and microlens array to generate uniform illumination. This method need not consider the light intensity of LED, but the manufacture of the microlens array is not easy. Moiseev and Doskolovich [12] first presented the analytical solution of total internal reflection (TIR) optical element to generate narrow-angle light distribution. The LED collimating lens design also has been developing constantly with the tool of freeform surface. Rubinstein and Wolansky [13] developed a method to design a freeform lens, which can redirect the light of a parallel beam to any prescribed light distribution. Chen and Lin [14] designed a freeform collimating lens and developed a modified freeform surface approximation method having more accuracy. Wang et al. [15] proposed a combination of a freeform lenscoupled parabolic reflector to solve the LED searchlight design problem. Wang et al. [16] designed two types of LED collimating lenses based on the theory of nonimaging optics and geometrical optics. Chen et al. [17] presented a freeform LED collimating lens method derived from basic geometric-optic, and this lens has better optical efficiency and smaller spot size compared with the lens in [13]. Aslanov et al. [18] proposed a compact optical system generating uniform irradiance distribution in a noncircular region. This optical system consists of a reflector and a microlens array. Using different shapes of the unit of a microlens array can generate any shape of illumination area. In an LED-based underwater optical detecting system, the nonuniformity of illumination causes the saturation on the center and dim on the edge. In order to overcome this © 2014 Optical Society of America
C. Chen and X. Zhang
Vol. 31, No. 5 / May 2014 / J. Opt. Soc. Am. A
shortcoming, the light pattern must be uniform and in a narrow view angle. In this paper, an optical system, which can collimate LED light uniformly, is proposed. This optical system consists of a TIR lens and a double-reflector system. The TIR lens collimates all the light of the LED parallel to the z axis but nonuniformly. Then Reflector 1 redirects the plane wavefront of the collimated light to uniform distribution, and Reflector 2 collimates the light again but uniformly. The simulation result shows this optical system can achieve illuminance uniformity more than 0.75, and the total output efficiency (TOE) of this system is 89.33%. We can control the radius of output light pattern through changing the radius of Reflector 2. In order to keep high uniformity and efficiency, the radius of Reflector 2 should not smaller than the radius of Reflector 1. This paper is organized as follows: Section 2 presents the design of the optical system. Section 3 presents the freeform construction process, which is based on the source–target energy map. Section 4 shows the simulation results and discusses some specific circumstances.
2. DESIGN OF THE OPTICAL SYSTEM Consider that the LED is a point source being collimated at a surface, which could be a refractive or a reflective one. In Fig. 1 light ray 1, 2, 3 refract at the optical surface and then parallel to the z axis. The light flux between light ray 1 and light ray 2 is equal to the light flux; therefore the light flux of the circle OR1 and the ring R1 R2 is equal. Because of the Lambertian light distribution of the LED, the angle θ1 is lager than θ2 . Hence the area of the circle is lager than the area of the ring R1 R2 , which causes unequal between the light distribution of the circle and the ring R1 R2 . So we can summarize that a single optical surface cannot collimate the light of the LED to a uniform light distribution. In order to achieve the uniform collimating of the LED, we propose the optical system demonstrated in Fig. 2. This optical system consists of three components: a TIR lens and two reflectors. The TIR lens collects and collimates all of the light of the LED to a plane but nonuniform wavefront. Reflector 1 redirects the nonuniform parallel light from the TIR lens to a uniform distribution on the reflector. Finally, Reflector 2 collimates the light to the uniform plane wavefront. Figure 3 shows the profile of the TIR lens, which has been used in [8]. This TIR lens has three freeform surfaces, which include two refractive and one reflective surface. Surface 1 directly refracts the light of the LED parallel to the z axis.
TIR
Reflector 1
Lens
Reflector 2 Fig. 2. Sketch map of the optical system.
But surface 1 cannot collimate all of the light of LED because of the TIR phenomenon, which can be overcome by Surface 2 and Surface 3. Surface 2 refracts the light to a bigger aspect angle, which guarantees that the TIR phenomenon will happen at Surface 3. Moreover the refractive light of Surface 2 can be regarded as the light emits from a point light source at f , which could be an arbitrary point at the negative x axis. The profile of the surface is a parabola whose focus point is point f . Thus the surface can reflect the light from Surface 2 parallel to the z axis. The points of surface can be calculated directly. Surface 1 and Surface 2 are freeform surfaces, which are constructed through the method presented in next section. As shown in Fig. 4, after being collimated by the TIR lens, the light of the LED is transformed to a plane wavefront, which is the input light beam of Reflector 1. The projection area of this wavefront at the plane z 0 is Ω. Then the light-intensity distribution of the input wavefront can be denoted as IΩ. The output wavefront of Reflector 2 is also a plane, and its projection area at the plane z 0 is T. Thus the light-intensity distribution of the output wavefront can be denoted as IT. The function IΩ is depended on the lightintensity distribution of the LED, and the function IT is a constant function, which is dependent on the whole light flux of the LED and the area of T. The calculation of IΩ will be shown in the next section, assuming all of the light of the input wavefront propagates to the output wavefront. For the conservation of energy, we can get the relationship between IΩ and IT as ZZ
R1
O
R2
1
ZZ IΩdΩ
L2
L1
ITdT:
(1)
z
2
θ1
3
3
θ2
1
2
x LED Fig. 1. Light of LED collimated by a single optical surface.
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f
LED
Fig. 3. Profile of the TIR lens.
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Reflector 2
The light-intensity distribution on a plane actually is illuminance, i.e., the total luminous flux incident on a surface per unit area. Denote E Ω and ET as the illuminance of the input and output wavefront. In the polar coordinate system, Eq. (1) changes into ZZ
E T r; θrdrdθ:
Z
ri
r i−1
0
Ω
(2)
Z E Ω r; θrdr dθ
2π
Z
Φ E T r; θrdr dθ : N r i−1
0
Output wave-front
In order to solve the shape data of the reflector numerically, the correspondence relationship between the points of the input and output wavefronts must be established. This correspondence relationship is based on the conservation of energy, i.e., Eq. (2). The input and output wavefronts are divided into same quantity of tiny patches, which have the same amount of light flux. Based on the edge principle [19], the light flux of the patches at the input wavefront will all propagate to the corresponding patches at the output wavefront if the edge rays of the patches at the input wavefront all propagate to the corresponding patches’ edge at the output wavefront. First we can obtain the relationship of the r in Eq. (2) of the input and output wavefront by doing integration along the r direction: 2π
wave-front
ZZ EΩ r; θrdrdθ
Z
Input
ri
T y
z A:
x
Z
ri r i−1
Z
θi
θi−1
Z E Ω r; θdθ rdr
ri r i−1
Z
Φ : E T r; θdθ rdr MN θi−1 θi
(4) In Eq. (4) M represents the number of patches along the θ direction in a ring. The edge points of these patches can be calculated through Eqs. (3) and (4). After that we can establish the one-to-one correspondence between the point of the input and output wavefronts.
3. FREEFORM SURFACE CONSTRUCTION A. Freeform Surface Construction of the TIR Lens Unlike Surface 3 in Fig. 3, a freeform surface could not be represented by an explicit function. To get a freeform surface, a large quantity of discrete points needs to be calculated one after another. For the TIR lens, we only need to calculate the profile and then revolve the profile around the z axis to get the shape of the lens. The reflectors are not symmetrical along the z axis; we need to calculate skeleton curves along the longitude direction then get the shape of the reflector by these curves. Figure 5 shows the process to construct freeform Surfaces 1 and 2 in Fig. 3, where the unit direction vector of the light ray emitting from the LED marked as I0 , I1 , I2 , and I3 hitting the freeform surface at P 0 , P 1 , P 2 , and P 3 , respectively, and O0 , O1 , O2 , and O3 are the unit direction vectors of the output light ray. In this figure, P 0 located at the z axis is the initial point,
Reflector 1 z
Fig. 4. Sketch map of the double reflector system.
which is set up according to the design require. The normal vector N0 at P 0 can be calculated by Snell’s law: 1 n2 − 2nO · I1∕2 N O − nI:
(5)
Then the tangent vector t0 at P 0 can be calculated by N0 . The next point of the lens profile P 1 is the point of intersection of the input light ray I1 and the tangent vector t0 . Assuming the angle between the light ray I1 and the z axis is θ1 , and the normal vector at P 0 is N0 N x ; N y . Then the coordinate of P 1 is
(3) In Eq. (3) Φ is the total luminous flux of the LED, and N is the number of patches along the r direction. Using Eq. (3) we get a series value of the radius. Then, doing the integration between two adjacent circles along the θ direction, we can get the relationship of θ between the input and output wavefronts:
=0
8
θth
7
Assuming the radius of the TIR lens is R then from Fig. 6, we can get the relationship of the radius of the TIR lens, and the radius of the input wavefront is
C. Chen and X. Zhang
Vol. 31, No. 5 / May 2014 / J. Opt. Soc. Am. A
O 0 O1 O 2
1121
1
O3 N3
N1 N 2
t2
t N0 t0 1
P3
P0 P1 P2
2
I 0 I1 I 2 I 3
LED Fig. 5. Construction process of the profile of the freeform surface of the TIR lens.
r R sin θ:
Using Eqs. (3), (4), (7), and (8), we can get the correspondence relationship between the points of input and output wavefronts. With this correspondence, the x; y coordinates of the two reflectors can be calculated. Figure 7 shows the construction process of the skeleton curves of the two reflectors. First choose the initial point of these reflectors, which are P 111 and P 211 . The superscripts 1 and 2 indicate which point belongs to which reflector, and the subscript 11 means this point is the first curve’s first point. The input vector at P 111 and output vector at P 211 are 0; 0; 1. The output vector at P 111 and input vector at P 211 are P211 − P111 ∕jP211 − P111 j. Then we can use Eq. (5) to calculate the normal vector at point P 111 and P 211 . Using these two normal vectors, we can calculate the tangent plane of these two points. The point of intersection between these two tangent planes and the next input light ray are the point P 112 and P 212 . Repeat this process until the last point of the curve is calculated. After the initial curves of the two reflectors are constructed, the rest of the skeleton curves of the reflector are calculated by the same procedure. First we calculate the normal vectors of the two initial curves. Then, using the tangent planes of the initial curves and the input light ray of next curve, we calculate the adjacent point of the next curve. The reflectors can be constructed by loft of the skeleton curves. According to the discussion in [8], the integrability of the source–target map affects the continuity of the freeform surface. In [5] the author believed that if there is a big discrepancy between the real normal vectors and the calculated Input wave-front of the reflectors
r
Fig. 7. Construct the skeleton curves of the reflectors.
(8)
rth
normal vectors, the irradiance on the target plane will differ from the prescribed irradiance. Figure 8 is the deviation between the real normal vectors and the calculated normal vectors of Reflector 1. The maximum value of the deviation is 5.6 × 10−3 , which is sufficiently small. Therefore we can construct the freeform surfaces of the reflector system without concern the discontinuity.
4. EXPERIMENT AND DISCUSSION After calculating the shape data of the TIR lens and the reflectors, the 3D model of the TIR lens and the reflectors were constructed in CAD software. Then the 3D models were imported to optical and illumination design software to validate the illumination pattern and the efficiency of the optical system. The CREE XR-E is used as the light source of this system. Figure 9(a) shows the 3D model of the TIR lens. We actually designed three reflector systems; each has a different radius of output reflector in order to generate different radius of light pattern. The radii of these reflectors are one times of TIR lens, -3
x 10 20
5
40 4
60 80
r
3
100 120
2 140 160
θth θr
3
1
180
1 θx
Fig. 6. Light propagates from the LED to the input wavefront.
200 50
100
150
200
250
300
0
Fig. 8. Deviation between the real normal vectors and the calculated normal vectors of Reflector 1.
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32.36mm
5mm
19.39mm
5mm 9.77mm
Reflector 1
Reflector 2
Fig. 9. Perspective view of the 3D model of the optical system. (a) TIR lens. (b) One time radius reflector.
1 0.9 0.8 0.7
Illuminance (lux)
two times of TIR lens, and 0.5 times of TIR lens. Figure 9 shows the 3D model of the one time radius reflector system. Figure 10(a) shows the light pattern after being collimated by the TIR lens, which is not as uniform as expected. Figure 10(b) shows the simulation result of the one time radius reflector system. The detecting surface is set to have the same x-y plane projective radius of Reflector 2 in Fig. 4. The uniformity of the light distribution at the detecting surface is defined as U E E min ∕Emax , in which E min and Emax represents the minimum and maximum illuminance of the detecting surface. From the relative illuminance distribution along the x and y axes of the detecting surface shown in Fig. 10(c), the uniformity across the x axis is U EX 0.78, and the uniformity across the y axis is U EY 0.82. As a comparison, the uniformity along the x and y axes of the TIR lens is U EX 0.0181 and U EY 0.0126. The TOE is defined as the ratio between the light flux of the output wavefront and the light flux of the LED. The ray trace result of Fig. 10(b) tells us that the TOE is 89%. Using the light efficiency of the TIR lens, we can calculate the light efficiency of the reflector system is 97%. Figure 11 shows the simulation result of two times radius reflector and one time radius reflector. The uniformity of two
0.6 0.5 0.4 0.3
X axis Y axis
0.2 0.1 0
-15
-10
-5
0
5
10
15
Distance to the center of the output plane (mm)
Fig. 10. (a) Light pattern collimated by TIR lens. (b) Light pattern after reshaping by the reflector system. (c) Illuminance along the x and y axes.
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0.1 0
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Fig. 11. Simulation result of different radius of reflector. (a) Two times radius reflector. (b) 0.5 times radius reflector.
0.6 0.5 0.4 0.3 0.2 X axis Y axis
0.1 0 -8
-6
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2
4
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Distance to the center of the output plane (mm)
Fig. 13. Simulation result of different radius of reflector using actual LED model. (a) Two times radius reflector. (b) One time radius reflector. (c) 0.5 times radius reflector.
source to simulate. The area of the light source in this Cree LED optical module is 1 mm × 1 mm. According to Fig. 8, the distance from the chip to Surface 2 of the TIR lens is less than five times the chip size. Therefore this actual Cree LED model could not be regarded as a point light source according to the far-field conditions of LEDs [20]. Figure 13 shows the 18000
0.5 times radius 1 times radius 2 times radius
0.5 times radius 1 times radius 2 times radius
14000
Intensity (Candle)
16000 14000
Intensity (Candle)
0.7
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12000 10000
0.8
Illuminance (lux)
times radius reflector in output plane along the x and y axes is U EX 0.76 and U EY 0.80. The TOE of two times reflector is 91%. From Fig. 11(b) we get the x and y axes uniformity of 0.5 times radius reflector being U EX 0.71 and U EY 0.63, the TOE of this reflector is 82%. The reason why the 0.5 times radius reflector has lower uniformity than the TOE is that the smaller radius of output reflector causes reduction of etendue of the system. In order to keep high uniformity and the TOE of the optical system, the radius of the output reflector should not smaller than the radius of input reflector. The luminous intensity distributions of these three optical systems are shown in Fig. 12. These three distribution curves almost overlap, which means the size of the reflector does not affect the luminous intensity distributions of the output beam. According to simulation results, the output beams of these optical systems have an FWHM of 2 × 2° . In order to investigate the optical performance of the design, we use the actual model of the Cree XR-E LED as the
0.9
FWHM = 2 × 2
8000 6000
12000 10000
FWHM = 2 × 4o
8000 6000 4000
4000
2000 2000 0 -30
0 -30 -20
-10
0
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30
Angle (degrees)
Fig. 12. Luminous intensity distributions of different optical systems.
-20
-10
0
10
20
30
Angle (degrees)
Fig. 14. Simulation results of the luminous intensity distributions of different optical systems using the actual LED model.
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Reflector2
TIR Lens
Reflector1
Fig. 15. Fabricated TIR lens and the two times radius reflectors.
simulations of these optical systems using the actual LED model. The uniformity of two times radius reflector in output plane along the x and y axes is U EX U EY 0.65. The TOE of two times reflector is 87%. The uniformity of the one time radius reflector in the output plane along x and y axes is U EX U EY 0.70. The TOE of two times reflector is 83%.
-30
-20
5. CONCLUSION
y(mm)
-10
0
10
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30 -30
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0 x (mm)
10
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1 -30
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0.7 -10 0.6 y(mm)
These uniformity and TOE are acceptable for the optical performance requirement. At the same time, the uniformity of the 0.5 times radius reflector in the output plane along the x and y axes is U EX U EY 0.59. The TOE of the 0.5 times radius reflector is 73%. The decline of the optical performance of the 0.5 times radius reflector is the worst among these reflectors. As per the discussion above, the reason for this is the decrease of the etendue. The actual model of the LED also affects the FWHM of the output beam. Figure 14 shows the luminous intensity distribution of these optical systems with the actual LED model. The maximum value of the 0.5 times radius reflector drops to 1.53 × 104 while the maximum values of the other two reflectors hardly have changed. Even though the FWHM of these three reflectors is nearly at the same value, which is 2 × 4°. As shown in Fig. 15, the TIR lens and the two times radius reflectors are fabricated by computer numerical control (CNC). The reflectors of this prototype are made of ABS, which was sputtered with an aluminum coating. The experimental result is shown in Fig. 16(a), and the pseudo color map of this result is shown in Fig. 16(b). Figure 16(b) shows clearly that there are some positioning errors during constructing the experimental setup. There are three main reasons: (1) the fabricated error of the CNC; (2) the coating error of the reflectors; (3) the installation error of the optical system. But, overall, we obtain a desirable design for the underwater optical-detecting system.
0
A single freeform surface lens cannot achieve uniform collimation of the LED light. In order to reach this goal, a double freeform reflector system is designed for this paper. This reflector system is designed by the source–target map method. First, the light of the LED is collimated by a TIR lens, and then the reflector system redistributes the light to a uniform light pattern and collimates the light again. The simulation result shows that the system can achieve 0.78 of uniformity and 89% TOE. We also investigate the optical performance of two other optical systems, which have different radii of the output reflector. The result shows that in order to keep high uniformity and light efficiency, the radius of the output reflector should not be smaller than the radius of the input reflector. Then, after simulation with an actual LED model, we discussed the impact of the size of the LED. Finally, we fabricated one of the designed optical systems. The experiment results show that the light pattern generated by this optical system satisfies our goal. The design of this paper can be applied in underwater photoelectric detecting systems such as underwater optical communications and underwater imaging.
0.5 0.4
10
0.3 20 0.2 30
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0 x (mm)
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Fig. 16. Results of the fabricated optical system. (a) Experimental result. (b) Pseudo color map of the experimental result.
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