Asymmetric Double Freeform Surface Lens for Integrated LED Automobile Headlamp

We propose a design method of asymmetric double freeform surface lens for an integrated LED automobile headlamp and develop an integrated LED automobile optical system. A single asymmetric double freeform surface lens is designed to redistribute rays emitting from the light source for realizing both low and high beams. Moreover, a freeform surface reflector is used to improve the energy efficiency of high beams. The prism placed in the optical path can suppress chromatic dispersion on the edge of the target plane. Simulation and experimental results show that the illumination values and color temperature of the key points can fully meet the requirements of United Nations Economic Commission for Europe vehicle regulations (ECE) R112, 48, and 128. The volume of the whole optical system comprised of freeform surface elements is smaller than that of the low beam system of a traditional headlamp, resulting in saved space, in which other electronic devices can be installed for the safety of the driver, which indicates that the proposed method is practical in the field of automobile lighting.


Introduction
LEDs have gradually entered our lives due to their advantages of energy saving, high energy efficiency, long lifespan, and miniaturization [1,2]. In the past decade, LEDs have been used increasingly as light sources in automobile headlamps for their excellent properties [3][4][5][6]. Since low-beam requirements are completely different from those of high beams, the traditional optical systems of low and high beams in headlamps are designed separately, mainly including projection and reflection types. In general, projection-type optical systems deflect by lenses and reflection type by reflectors [7]. It is easy to form a clear cutoff line but hard to improve the energy efficiency for projection types with a baffle plate; conversely, the reflection type without a baffle plate is more efficient but has a dim cutoff line. Previously, Cvetkovic et al. presented a simultaneous multiple surface (SMS) 3D method for automobile low and high beam headlamps, whose optical efficiency is more than 75% [8,9]. Domhardt et al. designed a combined lens for LED-based low beam headlamp but needed large space to install lenses for high energy efficiency [10]. Ge et al. used an elliptical and parabolic reflector to design a low beam headlamp that had high energy efficiency [5]. Hsieh et al. proposed a modular design for an LED vehicle projector headlamp that provided the cutoff line of low beams without a baffle plate [11]. Chu et al. proposed a low beam headlamp with a compound ellipsoidal reflector that achieved the highest energy efficiency in the existing literature [12]. In the past,

Design Method
The optical system of the new integrated headlamp is shown in Figure 1. The whole system consists of an ADFSL, a freeform surface reflector, a prism, a baffle, and two LEDs. The optical system of the low beam includes a LED source, an upper half of the ADFSL, and a baffle plate. Correspondingly, the optical system of the high beam comprises a freeform surface reflector, a prism, the lower half of the ADFSL, and a LED source. Rays emitting from the low beam LED source are partially blocked by the baffle plate and then redistributed by the ADFSL, forming a low beam with a clear cutoff line on the target plane. On the other hand, rays from the high beam LED source are focused on a point by the freeform surface reflector and then deflected by the prism. Finally, the remaining rays will be redistributed by the ADFSL to form the illumination distribution of the high beam. The implementation of the low and high beams will be described in detail below.

Optical System of the Low Beam Mode
Over the past decade, a series of regulations have been introd Low beam is mainly used for good road lighting and cannot cau drivers. According to the ECE R112 regulations, the tested point shown in Figure 2. It can be found that the light shape of the low beam has a wi both sides, which can provide a big angle of view for drivers. Mo of the low beam is "dark on the left but bright on the right". The in front of the headlamp for testing the illuminances on key poin has a specific function to ensure a qualified low beam on the meas the upper half of the ADFSL is used to realize the ellipse lighting blocks the stray light to form a clear cutoff line. In our laboratory laser headlamp without ellipsoid reflector (Figure 3) was design was also adopted in the integrated headlamp for the low beam fo

Optical System of the Low Beam Mode
Over the past decade, a series of regulations have been introduced for driver's safety. Low beam is mainly used for good road lighting and cannot cause glare to the opposing drivers. According to the ECE R112 regulations, the tested points and the cutoff line are shown in Figure 2.

Optical System of the Low Beam Mode
Over the past decade, a series of regulations have been introduc Low beam is mainly used for good road lighting and cannot cause drivers. According to the ECE R112 regulations, the tested points a shown in Figure 2. It can be found that the light shape of the low beam has a wide both sides, which can provide a big angle of view for drivers. Moreo of the low beam is "dark on the left but bright on the right". The me in front of the headlamp for testing the illuminances on key points. has a specific function to ensure a qualified low beam on the measuri the upper half of the ADFSL is used to realize the ellipse lighting spo blocks the stray light to form a clear cutoff line. In our laboratory, a laser headlamp without ellipsoid reflector ( Figure 3) was designed, was also adopted in the integrated headlamp for the low beam for th optical system. It can be found that the light shape of the low beam has a wide horizontal spread for both sides, which can provide a big angle of view for drivers. Moreover, the lighting spot of the low beam is "dark on the left but bright on the right". The measuring plane is 25 m in front of the headlamp for testing the illuminances on key points. Each optical element has a specific function to ensure a qualified low beam on the measuring plane. Concretely, the upper half of the ADFSL is used to realize the ellipse lighting spot, and the baffle plate blocks the stray light to form a clear cutoff line. In our laboratory, a low beam system for laser headlamp without ellipsoid reflector ( Figure 3) was designed, and the construction was also adopted in the integrated headlamp for the low beam for the compactness of the optical system. blocks the stray light to form a clear cutoff line. In our laboratory laser headlamp without ellipsoid reflector (Figure 3) was design was also adopted in the integrated headlamp for the low beam fo optical system.  Double freeform surface lens (DFSL) has many advantages over single freeform surface lens. For example, it can achieve an accurate distribution of light and suppress chromatic dispersion by its inner and outer freeform surfaces [15]. The deflection factor C (0 ≤ C ≤ 1), an important factor in DFSL, is used to represent the angular deflection ratio of two freeform surfaces. It is worth noting that θ, ϕ are zenith angle and azimuth angle, respectively, and α is the angle between the discrete points on the target plane and positive X axis ( Figure 4). The ray's angle after passing through the inner freeform surface could be denoted as follows: omachines 2021, 12,663 Double freeform surface lens (DFSL) has many advantages ov face lens. For example, it can achieve an accurate distribution of lig matic dispersion by its inner and outer freeform surfaces [15]. The d C ≤ 1), an important factor in DFSL, is used to represent the angular freeform surfaces. It is worth noting that θ , ϕ are zenith angle a spectively, and α is the angle between the discrete points on the tive X axis ( Figure 4). The ray's angle after passing through the i could be denoted as follows: (1 ) To construct the ADFSL, the mapping relationship between ce target plane and discrete points of the target plane should be calcu the division of the target illumination area. To simplify the calculat ified illumination distribution, the lighting zone is set as an ellipse To construct the ADFSL, the mapping relationship between certain angle rays on the target plane and discrete points of the target plane should be calculated. Figure 5 shows the division of the target illumination area. To simplify the calculation and generate qualified illumination distribution, the lighting zone is set as an ellipse and divided into I, II, III zones. E1, E2, E3 are the corresponding illuminances of zone I, II, III, respectively, and the semi-major axis is a, and the semi-minor axis, b. Similarly, the semi-major and semi-minor axes are equally divided into three parts, with a i , b i being the semi-major axis and semi-minor axis of the divided sub-ellipse, respectively. On the other hand, the target illumination area is divided into n parts along the direction α, where α i is the angle between ith part and the positive X axis. III zones. E1, E2, E3 are the corresponding illuminances of zone I, II, I the semi-major axis is a, and the semi-minor axis, b. Similarly, the se minor axes are equally divided into three parts, with i a , i b being and semi-minor axis of the divided sub-ellipse, respectively. On the ot illumination area is divided into n parts along the direction α , wh between ith part and the positive X axis.  I 0 is the central intensity of the LED source; I (θ) , the intensity distribution of light in the θ direction; θ, the angle between the emergent light and the optical axis; and the relationship between I (θ) and I 0 can be expressed as follows: According to the conservation of energy, we obtain [30] 2π By solving the above integral equation, the mapping relationship between θ i and the illumination area on the target plane can be acquired.
According to the conservation of energy, the relationship between the azimuth angle ϕ j and α j can be expressed as follows: By solving the above equation, the relationship between ϕ j and α j is obtained.
The coordinates of the points on the target plane can be calculated by Equation (5). Then, we obtain the mapping relationship between the light emitting from the source and the points on the target plane. Next, the profile of ADFL is calculated by iteration. The initial points of the first and second freeform surfaces are set as L1 and L2, respectively.
After the initial points are given, the normal vector of the initial points can be obtained according to the vector form of refraction law, In is the unit vector of the incident ray, → Out is the unit vector of outgoing ray, → N is the normal vector, n o is the refractive index in the outgoing space, and n in is the refractive index in the incident space. All coordinates of the discrete points on the freeform surface can be calculated by iteration of the normal vector, which stipulates that the intersection point of the next incident ray and the normal plane of the last point is the next point on the curve. The expression of the normal plane at any point can be obtained by Equation (6). By fixing the value of ϕ and changing the value of θ, the initial curve of the first and second freeform surfaces is generated. Then, by fixing the value of θ and changing the value of ϕ, the coordinates of the discrete points on both freeform surfaces can be acquired by iteration of the normal vector. Readers can find the detailed iteration process in [15]. After importing the data to modeling software, the upper half of the ADFSL can be obtained by lofting curve, as shown in Figure 6.
omachines 2021, 12,663 the first and second freeform surfaces is generated. Then, by fixing th changing the value of ϕ , the coordinates of the discrete points on both can be acquired by iteration of the normal vector. Readers can find th process in reference 15. After importing the data to modeling software the ADFSL can be obtained by lofting curve, as shown in Figure 6.

Optical System of the High Beam Mode
The high beam of the automobile headlamp is mainly used for bad Micromachines 2021, 12,663 the first and second freeform surfaces is generated. Then, by fixing the va changing the value of ϕ , the coordinates of the discrete points on both fre can be acquired by iteration of the normal vector. Readers can find the de process in reference 15. After importing the data to modeling software, th the ADFSL can be obtained by lofting curve, as shown in Figure 6.

Optical System of the High Beam Mode
The high beam of the automobile headlamp is mainly used for bad road it provides enough illumination on the target area [31]. In addition, the regu the target plane center area with maximum illumination. The test area of th shown in Figure 8.

Optical System of the High Beam Mode
The high beam of the automobile headlamp is mainly used for bad road lighting since it provides enough illumination on the target area [31]. In addition, the regulations require

Optical System of the High Beam Mode
The high beam of the automobile headlamp is mainly used for ba it provides enough illumination on the target area [31]. In addition, the the target plane center area with maximum illumination. The test area shown in Figure 8. The traditional design uses an ellipsoidal reflector to focus the ra source, while part of the rays emitted from the high beam source wi low beam source due to the volume of the LED source. It will caus losses, as shown in Figure 9a. Therefore, the construction in Figure 1 is light efficiency and illumination. Rays emitted from the high beam sou on Z axis by the freeform surface reflector, as shown in Figure 9b. The traditional design uses an ellipsoidal reflector to focus the rays emitted from the source, while part of the rays emitted from the high beam source will be blocked by the low beam source due to the volume of the LED source. It will cause significant energy losses, as shown in Figure 9a. Therefore, the construction in Figure 1 is adopted to improve light efficiency and illumination. Rays emitted from the high beam source focus on a point on Z axis by the freeform surface reflector, as shown in Figure 9b. is the normal vector. The initial curve and profile of the freeform surface refl can be obtained by an iteration process.
Freeform surface reflector converges rays emitted from the source to another f on the Z axis. Therefore, the focus on the Z axis can be seen as a light source wit volume that does not block the rays from the low beam source and will illuminat target plane. Similar to the process of low beam design, the lighting spot of the high b is set as an ellipse, whose semi-major axis and semi-minor axis are equally divided in parts. Accordingly, i a , i b are the semi-major and semi-minor axes of ith part, re Due to the symmetry of the reflector, only half of the freeform surface reflector was designed. An initial point is given and the normal vector of the initial point can be obtained according to the vector form of the reflection law, which can be expressed as where → In, → Out are the unit vectors of the incident ray and outgoing ray, respectively, and → N is the normal vector. The initial curve and profile of the freeform surface reflector can be obtained by an iteration process.
Freeform surface reflector converges rays emitted from the source to another focus on the Z axis. Therefore, the focus on the Z axis can be seen as a light source without volume that does not block the rays from the low beam source and will illuminate the target plane. Similar to the process of low beam design, the lighting spot of the high beam is set as an ellipse, whose semi-major axis and semi-minor axis are equally divided into m parts. Accordingly, a i , b i are the semi-major and semi-minor axes of ith part, respectively, Micromachines 2021, 12, 663 8 of 16 0 ≤ i ≤ m. At the same time, the target plane is also divided into n parts along the α direction, α i is the angle between ith part and the X axis, and then the target plane is divided into m × n lattices, as shown in Figure 10. According to the energy conservation, we can obtain the mapping relations.
where E 0 ·k i is the illumination value, and k i is the illuminance control factor on the ith part. Generally, the closer to the original point, the greater the value of the k i . I(θ, ϕ) is the intensity distribution of the light focused by the freeform surface reflector. After calculating Equation (8), the mapping relationship can be acquired, and the profile of the lower half of the ADFSL is calculated by the normal vector iteration.
where In   , Out  are the unit vectors of the incident ray and outgoin and N   is the normal vector. The initial curve and profile of the freefo can be obtained by an iteration process.
Freeform surface reflector converges rays emitted from the sou on the Z axis. Therefore, the focus on the Z axis can be seen as a l volume that does not block the rays from the low beam source and target plane. Similar to the process of low beam design, the lighting sp is set as an ellipse, whose semi-major axis and semi-minor axis are eq parts. Accordingly, i a , i b are the semi-major and semi-minor axes tively, 0 ≤ i ≤ m. At the same time, the target plane is also divided into direction, i α is the angle between ith part and the X axis, and then divided into m n × lattices, as shown in Figure 10. According to the e we can obtain the mapping relations.
where E0•ki is the illumination value, and ki is the illuminance control f Generally, the closer to the original point, the greater the value of th intensity distribution of the light focused by the freeform surface refle ing Equation (8), the mapping relationship can be acquired, and the half of the ADFSL is calculated by the normal vector iteration. Similar to the low beam lens construction method, the whole ADFSL can be built in the modeling software. As shown in Figure 11, the diameter and thickness of the lens are 64 mm and 30 mm, respectively. sin( α + δ 2 ) = n sin( α 2 )· cos I 1 +I 2 2 cos I 1 +I 2 2 (9) Micromachines 2021, 12, 663 8 of 16 Similar to the low beam lens construction method, the whole ADFSL can be built in the modeling software. As shown in Figure 11, the diameter and thickness of the lens are 64 mm and 30 mm, respectively.  Although DFSL can reduce chromatic dispersion on the target plane, there is still serious chromatic dispersion at the edge due to the large deflection angle. To obtain a light pattern with a stable color distribution, a prism was added to the optical system to reduce the deflection angle of the ADFSL. Figure 12 shows an optical path in the prism. The prism can deflect rays based on refraction law, and the deflection angle between the incident and outgoing ray is δ. Formula (9) presents the calculation of the deflection angle of the prism [32]. Although DFSL can reduce chromatic dispersion on the targ serious chromatic dispersion at the edge due to the large deflection a pattern with a stable color distribution, a prism was added to the op the deflection angle of the ADFSL. Figure 12 shows an optical path in can deflect rays based on refraction law, and the deflection angle and outgoing ray is δ . Formula (9) presents the calculation of the d prism [32].

Simulation
After calculation and optimization, the whole system of the in introduced into the optical simulation software LightTools [33], as The volume of the whole system is 91 × 64 × 64 mm 3 , which is sma rately designed headlamp systems. The low beam source placed H4L531.TE of OSRAM company and 6R brightness with 1300 lm lum [34]. On the other hand, the high beam source placed at a focus o reflector is KW H4L531.TE and 7R brightness with 1500 lm luminou and high beam sources comply with the ECE R48 and ECE R128 reg

Simulation
After calculation and optimization, the whole system of the integrated headlamp is introduced into the optical simulation software LightTools [33], as shown in Figure 13. The volume of the whole system is 91 × 64 × 64 mm 3 , which is smaller than all the separately designed headlamp systems. The low beam source placed at the origin is KW H4L531.TE of OSRAM company and 6R brightness with 1300 lm luminous flux is selected [34]. On the other hand, the high beam source placed at a focus of the freeform surface reflector is KW H4L531.TE and 7R brightness with 1500 lm luminous flux is selected. Low and high beam sources comply with the ECE R48 and ECE R128 regulations [28,29]. Before the simulation, the freeform surface reflector is set as an aluminum all a reflectivity of 90% and the lens material is PMMA with a refractive index of 1 baffle plate is set as a perfect absorber to absorb all rays that hit it. The exact illum values on key points and the specified regions in the prescribed area can be rea software. The simulation results are shown in the figures and tables below. Figures 14 and 15 show the illumination and color distribution of the low b Before the simulation, the freeform surface reflector is set as an aluminum alloy with a reflectivity of 90% and the lens material is PMMA with a refractive index of 1.49. The baffle plate is set as a perfect absorber to absorb all rays that hit it. The exact illuminance values on key points and the specified regions in the prescribed area can be read in the software. The simulation results are shown in the figures and tables below. Figures 14 and 15 show the illumination and color distribution of the low beam, respectively. Figure 16 shows the illumination distribution of the high beam. Figure 17a shows the color distribution of a high beam with a prism, and Figure 17b shows the color distribution without a prism. Tables 1 and 2 show the simulated illuminance values on the target plane of low and high beams, respectively. Before the simulation, the freeform surface reflector is set as an al a reflectivity of 90% and the lens material is PMMA with a refractiv baffle plate is set as a perfect absorber to absorb all rays that hit it. Th values on key points and the specified regions in the prescribed area software. The simulation results are shown in the figures and tables b Figures 14 and 15 show the illumination and color distribution o spectively. Figure 16 shows the illumination distribution of the hig shows the color distribution of a high beam with a prism, and Figure distribution without a prism. Tables 1 and 2 show the simulated illu the target plane of low and high beams, respectively.    Before the simulation, the freeform surface reflector is set as an al a reflectivity of 90% and the lens material is PMMA with a refractiv baffle plate is set as a perfect absorber to absorb all rays that hit it. Th values on key points and the specified regions in the prescribed area software. The simulation results are shown in the figures and tables b Figures 14 and 15 show the illumination and color distribution o spectively. Figure 16 shows the illumination distribution of the hig shows the color distribution of a high beam with a prism, and Figure distribution without a prism. Tables 1 and 2 show the simulated illu the target plane of low and high beams, respectively.

25R
≥2 Zone I ≤2E * Zone III ≤0.7 Zone IV ≥3 E * is the actual measurement of the value at point 50R.       Figures 14 and 15 indicate that the integrated headlamp system with a single ADFSL could ensure the quality of the low beam. The illumination width of the low beam can reach 12 m, which can provide a large angle of view for drivers. Furthermore, the cutoff line with the blue edge does not appear, which means that the inherent problem of the cutoff line with the blue edge in traditional projection designs can be solved by this method. The color temperature of the low beam is stable on the target plane, and the color on the target plane is close to white. As shown in Table 1, the simulation values of the low beam fully meet the requirements of ECE R112 regulations; Figure 16 shows that the illumination distribution of the high beam can comply with the requirements of ECE R112 regulations. Furthermore, the highest value of the high beam is 80.8 lux, providing sufficient illumination when the road lighting is poor. Figure 17 shows the color distribution of the high beam. The color distribution with a prism is more stable, as compared to the color distribution without a prism, which means that adding a prism to the optical path can largely suppress chromatic dispersion on the edge. Table 2 demonstrates that the simulation values of the high beam satisfy the requirements of ECE R112 regulations. In short, the light shape of the low beam and high beam and simulation values of the key points fully comply with the requirements of ECE R112; Moreover, the chromatic dispersion can be largely suppressed by using a new integrated optical system.
Here, we must present a simple analysis of the installation error. The error of the low beam and high beam source position mounting is ±0.3 mm and 0.5 mm, respectively. In other words, when position error is greater than the above values, the performance will no longer satisfy the ECE R112 regulations. Therefore, proper installment of the optical elements is very important for achieving optimal performance.

Experiment
To study the illumination performance of the experiment, the optical samples were processed and assembled according to the results of the optical simulation software, and the GO-HD5 automobile lighting test system was used for testing the illuminance values of the key points and specified regions. According to ECE R112 regulations, the test starts after the light source is steady. The lighting spot of the low beam and color temperature test of the point 50 V are shown in Figure 18, and the lighting spot of the high beam and color temperature test of the point HV are shown in Figure 19. The test illumination values of the low beam and high beam measurement screens are shown in Tables 3 and 4, respectively.
From Figure 18a, there is a straight cutoff line with no stray light above, and the color temperature is stable on the measuring screen. Slight chromatic dispersion appears near the cutoff line, which is different from the simulation results. Additionally, it still meets the ECE regulations. Compared with the traditional low beam, the chromatic dispersion near the cutoff line is much slighter. The test results of illuminance were slightly different from the simulations, due to the slight error of the shield's place and the absorption of mechanical objects. However, all illuminance values at the required points can fit with the regulations. According to ECE regulations, the color test results for the 50 V point must be within the white light range. The color test result at point 50 V is 5160 K, as shown in Figure 18b, and is quite qualified. The low beam lighting spot has a smooth transition that provides drivers with visual comfort.       It can be inferred from Figure 19a and Table 4 that the lighting spot is nearly elliptic and the central area is the brightest, which meets the ECE regulations. The color of the high beam is almost white, indicating that adding a prism could reduce chromatic dispersion at the edge of the target area and stabilize the color temperature. Similar to the test low beam results, the high beam test results in Table 4 are lower than the simulation results. The maximum illuminance of the high beam is 70.36 lux, less than simulation results but still much greater than the regulation requires, and can provide sufficient illumination for the ground. Similarly, the color test results for the HV point must be within the white light range. The color test result of the HV point is 5560 K, as shown in Figure 19b, which complies with the regulations.

Discussion
According to the simulation and experimental results, it can be found that both illuminance values of the required points and the color temperature distributions are qualified. The comparison of the new integrated headlamp system with the traditional headlamp system is shown in Table 5. The volume of the new integrated LED headlamp is 91 × 64 × 64 mm 3 , much smaller than that of the traditional headlamp. The ADFSL has a diameter of 64 mm, as large as the traditional low beam lens, which means saving space for installing the high beam system and its cooling system. Compared with the high beam without prism, the new headlamp with ADFSL and prism has a more uniform color distribution due to the light deflection of the prim. Both low and high beams are qualified. Since the new integrated headlamp has only one optical system, it costs less than the traditional headlamp with two optical systems. By using an asymmetric double freeform surface lens, the cutoff line of the new integrated headlamp has less chromatic dispersion than that of the traditional headlamp using a single freeform surface or aspheric surface lens, providing comfortable lighting for oncoming drivers. These advantages make the new integrated headlights acceptable in the headlamp market.

Conclusions
We proposed a method to design the asymmetric double freeform surface lens and the integrated LED headlamp optical system. The construction algorithms of the optical elements were presented in detail. Simulation and experimental results show that a good light pattern can be obtained with the selected source, and the values of key points and specified regions can fully comply with the standards of ECE R112, 48, and 128. Moreover, color uniformity on the target plane is improved by the freeform surface lens and the prism. The volume of the whole optical system is smaller than that of the traditional headlamp, saving space to install other electronic devices safely, such as distance sensors, to help drivers gain better control of the distances between vehicles. Moreover, the freeform surface optical system has high compactness and manufacturing feasibility.