Home > Knowledge > Content

Product Categories

Contact Information

  • Wuxi Kinglux Glass Lens Co.Ltd
  • ADD:No.286,Changjiang North Rd,New Dist,Wuxi,JS prov,China
  • Tel: 86-510-66759801
  • Fax: 86-510-84602998
  • Mobile phone: 86-18168862789
  • E-mail: ledglasslens@163.com
  • Contact person: Huimin Zhang
  • Development of multiple-surface optical elements for road lighting
    May 17, 2018

    The development of LED secondary optics for road illumination is quite a challenging problem. Optical elements developed for this kind of application should have maximal efficiency, provide high luminance and illuminance uniformity, and meet many other specific requirements. Here, we demonstrate that the usage of the supporting quadric method modification enables generating free-form optical solution satisfying all these requirements perfectly. As an example, two optical elements for different roadway types are computed, manufactured by injection molding, and then measured in a photometry bench. Experimental data demonstrate that the obtained light distributions meet ME1 class requirements of EN 13201 standard. The obtained directivity patterns are universal and provide high performance with different configurations of luminaires’ arrangement: the ratio of pole altitude to distance can vary from 2.5 up to 3.6.


    © 2017 Optical Society of America

    1. Introduction

    Due to the presence of many obvious advantages, nowadays light-emitting diodes (LEDs) leave behind incandescent and fluorescent light sources. The usage of LEDs in different lighting devices requires the application of so-called secondary optics. Secondary optics is the class of optical elements redirecting light from the source to the prescribed illuminated area.

    One of the greatest markets of LEDs application is road and street lighting. The problem of designing the optical element providing appropriate photometry for road lighting is quite challenging [1–5], because it is necessary to consider the properties of the roadway surface, the distance between light poles, the number and geometry of traffic lanes, etc. In [1, 2], methods for computing optical elements with specific inner surfaces, working on the TIR-principle (total internal reflection), are proposed. Unfortunately, the authors haven’t presented the obtained light performance characteristics (such as luminance and illuminance uniformity, “Threshold Increment”, etc.) that are the most important criteria in road lighting. The Section 3 of [3] considers an approach intended for computation of a perfect intensity distribution for street-lighting application, then in the Section 4 of [3] with usage of method [4] an optical element with piecewise smooth surface, generating such a distribution, is designed. Such a surface usually causes problems in manufacturing the optical element due to the limitations of the injection molding method.

    The most popular design of the LED optical element for road lighting contains two working smooth refractive surfaces [6–12]. This concept enables low Fresnel losses and allows an intensity distribution to be generated with any angular size required for this kind of problems. In our recent paper [12], the modification of the supporting quadrics method (SQM) [13–15] for computation of the optical element with two smooth surfaces for accurate generation of the any required asymmetrical intensity distribution is presented. Here, we demonstrate the applicability of SQM modification [12] to the road lighting problem. Two smooth and efficient optical elements, generating symmetrical and asymmetrical intensity patterns respectively, are computed and then manufactured by injection molding technology. The measured photometry of both optical elements has passed all ME1 class tests for the considered road configurations successfully.

    2. Method for designing optical element with multiple free-form surfaces

    Single optical elements are almost not utilized in road lighting applications since the use of multi-lens arrays (multilenses) is cheaper, more reliable and efficient. The multilens is a plastic base with several single optical elements embedded. Such an optical element is usually mounted on a printed circuit board (PCB) with an LED array (Fig. 1). In this case, each single lens transforms the emitted light from a certain LED; therefore, the problem of designing a multi-lens array is reduced to the computation of single optical elements. For the purpose of description completeness, a brief solution of this problem, according to [12], is presented below.


    Fig. 1 Q1.jpg

    Arrangement of PCB, LEDs and multi-lens array.


    2.1 Statement of the problem

    Let us assume that the point light source (LED) with prescribed intensity distribution 𝐼𝑠𝑟𝑐(𝑠0)Isrc(s0)is located at the origin of coordinates, where 𝐬0s0 is a unit vector. An optical element with refractive index 𝑛n covers the LED, thus all emitted rays are refracted by the inner and outer surfaces sequentially, as it is shown in Fig. 2. We need to compute the inner and the outer surfaces to generate the required intensity distribution 𝐼𝑟𝑒𝑞(𝑥)Ireq(x) depending on the direction unit vector x.


    Fig. 2 Q2.jpg

    Arrangement of light source and optical element.


    2.2 Inner surface computation

    The optical element surfaces (i.e. inner and outer) are designed sequentially. For each surface, a piecewise smooth solution is computed, then the NURBS-spline is fitted on it, providing a smooth surface.

    For computing a piecewise smooth solution, we propose to apply the supporting quadric method [13–15]. Let us approximate the required continuous intensity distribution 𝐼𝑟𝑒𝑞(𝑥)Ireq(x) by discrete light distribution (𝑥𝑖,𝛷𝑖),𝑖=1,...,𝑁(xi,Φi), i=1,...,N, which corresponds to the set of collimated light beams with fluxes 𝛷𝑖Φi, propagating in the directions 𝑥𝑖xi. The range of definition of function 𝐼𝑟𝑒𝑞(𝑥)Ireq(x) should be divided into small solid angles 𝛿𝛺𝑖δΩi, and after that the values 𝛷𝑖Φi and 𝑥𝑖xi are computed as the integrated flux and the weighted direction in the solid angles 𝛿𝛺𝑖δΩi, respectively:

    (1)

    𝛷𝑖=𝛿𝛺𝑖𝐼(𝑥)d𝛺,Φi=∫δΩiI(x)dΩ,

    (2)

    𝑥𝑖=𝛿𝛺𝑖𝑥𝐼(𝑥)d𝛺𝛷𝑖.xi=∫δΩixI(x)dΩΦi.

    The piecewise smooth inner surface 𝐫𝐂0rC0 is defined as a set of 𝑁N segments. Each segment narrows the angular size of the incident beam, making the bundle more “collimated” in the direction 𝑥𝑖xi . This means that the following equality is implemented:

    (3)

    𝛾=𝑘𝜓,γ=kψ,

    where 𝜓ψ and 𝛾γ are the angles between the ray and the direction 𝐱𝑖xi before and after refraction on the inner surface segment, respectively (Fig. 3). Each 𝑖i-th single segment is a part of axisymmetrical surface with symmetry axis 𝐱𝑖xi and with profile defined by the following explicit expression as it was shown in [12]:

    (4)

    𝑟𝑖(𝐬0)=𝐬0𝑟0𝑖(𝑛cos((1𝑘)arccos(𝐬0𝑥𝑖))1𝑛1)1𝑘1,ri(s0)=s0r0i(ncos((1−k)arccos(s0⋅xi))−1n−1)1k−1,

    where 𝐬0s0 is an arbitrary unit vector, 𝑟0𝑖r0i is a scalar parameter defining the size of 𝑖i -th segment, and 𝑛n is a refractive index of the lens material.



    Fig. 3 Q3.jpg

    Passing rays through the segment of the inner surface.


    Coefficient 𝑘[0;1]k∈[0;1] determines the part of work on the ray deflection, performed by the inner surface. It can be easily shown from the Eq. (4), that in the limit case of 𝑘=0k=0 the inner segment becomes a hyperboloid, completely collimating the incident bundle in the direction 𝐱𝑖xi; and in the case of 𝑘=1k=1 the shape of the segment is spherical, that does not deflect rays at all.

    The entire surface 𝐫𝐂0rC0, consisting of segments defined by Eq. (4), is described as follows [12]:

    (5)

    𝐫𝐂0(𝐬0)=𝑟𝑖(𝐬0)𝐬0,𝑖=argmin𝑗{1,...,𝑁}𝑟𝑗(𝐬0),𝑟𝑗(𝐬0)=𝑟0𝑗(𝑛cos((1𝑘)arccos(𝐬0𝑥𝑗))1𝑛1)1𝑘1.{rC0(s0)=ri(s0)s0,i=argminj∈{1,...,N}rj(s0),rj(s0)=r0j(ncos((1−k)arccos(s0⋅xj))−1n−1)1k−1.

    The shape of the piecewise surface (Eq. (5)) is completely defined by the set of parameters 𝑟0𝑖,𝑖=1,...,𝑁r0i,i=1,...,N. Each parameter specifies the size of the 𝑖i-th segment and the flux 𝛷𝑖Φi, propagating in direction 𝑥𝑖xi. For parameter 𝑟0𝑖r0i computation, the supporting quadric method is used [15]. In this case, computation of the piecewise smooth inner surface, producing the required discrete light distribution, is reduced to the following action sequence [12]:

    • 1. Presetting vector of parameters with initial values 𝑟0𝑖r0i.

    • 2. Computation of discrete light distribution 𝛷𝑖,𝑐𝑎𝑙𝑐Φi,calc, approximating the required continuous distribution 𝐼𝑟𝑒𝑞(𝑥)Ireq(x). We assume that all rays passed through the 𝑖i-th segment and partially deflected to the 𝐱𝑖xi direction, then are going to be collimated by the outer surface in the same direction. In order to compute 𝛷𝑖,𝑐𝑎𝑙𝑐Φi,calc, a special ray-tracing procedure should be performed: for each ray 𝐬0s0 the values 𝑟𝑗(𝐬0),𝑗=1,...,𝑁rj(s0),j=1,...,Nare computed using Eq. (4); then the index 𝑖i of minimal radius-vector 𝑟𝑗rj should be found, and the energy of the ray 𝐬0s0 is added to the 𝑖i-th point of discrete light distribution.

    • 3. The maximum error max(|𝛷𝑖,𝑐𝑎𝑙𝑐𝛷𝑖|)max(|Φi,calc−Φ i|) calculation, where 𝛷𝑖Φ i is the required flux value in the 𝐱𝑖xi direction. If the current error is more than the prescribed accuracy, Step 4 should be followed; otherwise, the design of inner piecewise smooth surface has finished.

    • 4. Adjusting the values 𝑟0𝑖=𝑟0𝑖±𝛥𝑖,𝑟r0i =r0i ±Δi,r, and returning to Step 2. 𝛥𝑖,𝑟Δi,r is a positive constant. Quantity 𝑟0𝑖r0i defines the distance from the source to the surface segment. As follows from the Eq. (5), increasing 𝑟0𝑖r0i leads to the reducing the light flux, which is incident on the 𝑖i-th segment and, finally, is collimated in the direction 𝐱𝑖xi. Otherwise, the decrease of value 𝑟0𝑖r0i provides more flux in the corresponding direction. In [14, 15], a simple way of segment parameters adjustment is proposed. The described method guarantees the convergence of this iteration process and can be utilized to solve this problem.

    This algorithm makes it possible to obtain a piecewise smooth surface, performing a fraction of the work of the ray deflection equal to 1𝑘1−k, to generate the required discrete intensity distribution (𝑥𝑖,𝛷𝑖),𝑖=1,...,𝑁(xi,Φi), i=1,...,N. The smooth inner surface is obtained by fitting a NURBS spline on the resulting surface 𝐫𝐂0(𝐬0)rC0(s0).

    2.3. Outer surface computation

    The outer surface can be computed in a similar way: in the beginning, a piecewise smooth surface, generating a discrete intensity distribution consisting of 𝑁N primitives, should be designed; then the obtained solution will be fitted by a NURBS spline. In the case of the outer surface, the base segment collimates incident light flux in the corresponding direction 𝐱𝑖xi (Fig. 4).


    Fig. 4 Q4.jpg

    Passing rays through the segment of the outer surface.


    As in the case of the inner surface, the 𝑖i-th outer surface segment 𝑀𝑖(𝐬0)Mi(s0) is a part of axisymmetrical surface with symmetry axis 𝐱𝑖xi, its shape can be computed with the Fermat principle [16] as it was shown in [12]:

    (6)

    𝑀𝑖(𝐬0)=𝑟(𝐬0)+𝑙𝑖(𝐬0)𝐬1,𝑙𝑖(𝐬0)=𝑙0𝑖(1𝑛)+𝑟(𝐬0)(1(𝐬0𝑥𝑖))(𝐬1(𝐬0)𝑥𝑖)𝑛,{Mi(s0)=r(s0)+li(s0)s1,li(s0)=l0i(1−n)+r(s0)(1−(s0⋅xi))(s1(s0)⋅xi)−n,

    where 𝐫(𝐬0)r(s0) is the smooth inner surface, 𝑙𝑖(𝐬0)li(s0) is a scalar function, defining the distance between the inner surface 𝐫(𝐬0)r(s0) and the point 𝐌𝑖(𝐬0)Mi(s0) of the outer surface along the refracted ray 𝐬1(𝐬0)s1(s0) within the optical element; 𝑙0𝑖l0i is the parameter of the 𝑖i-th segment. The entire piecewise outer surface is determined similar to the inner one as follows (the particular description of the outer surface computation is presented in [12]):

    (7)

    𝑀(𝐬0)=𝑟(𝐬0)+𝑙𝑖(𝐬0)𝐬1,𝑖=argmin𝑗{1,...,𝑁}𝑙𝑗(𝐬0),𝑙𝑗(𝐬0)=𝑙0𝑗(1𝑛)+𝑟(𝐬0)(1(𝐬0𝑥𝑗))(𝐬1(𝐬0)𝑥𝑗)𝑛,{M(s0)=r(s0)+li(s0)s1,i=argminj∈{1,...,N}lj(s0),lj(s0)=l0j(1−n)+r(s0)(1−(s0⋅xj))(s1(s0)⋅xj)−n,

    where 𝐌(𝐬0)M(s0) is vector function of the piecewise outer surface.


    As in the case of the inner surface, the set of parameters 𝑙0𝑗l0j completely defines the shape of the outer surface. It can be computed with use of the algorithm described above in the Section 2.2.

    3. Developed examples

    Since the general goal of road lighting is to provide clear visibility and an absence of glaring effect at night, there are many requirements to be met. All the particulars regarding road illumination are described in EN 13201 Standard for European countries and in IESNA Standard for Northern America. According to these rules, the main quality criteria are the parameters that are based on average luminance and illuminance uniformity, together with a maximum “Threshold Increment”, which is a parameter designed to control glare and a “Surround Ratio”. The latter controls the minimum luminance of a strip adjacent to the carriageway when there are no traffic areas with their own requirements adjacent to the carriageway. A clear and detailed description of each parameter is also presented in [3].

    Here we propose to develop two multilenses: the first will generate light distribution for luminaire, mounted in the median strip of motorway with two traffic lanes in each direction; and the second will form the distribution for the poles, arranged on the sides of the motorway. Both examples will meet the requirement for the highest lighting class ME1 of EN 13201 Standard.

    The design of the multilens for road illumination includes the following steps:

    • 1. Generation of the prescribed continuous photometry meeting the current roadway requirements. The solution of this problem is clearly described in [3].

    • 2. Discretization of the prescribed photometry using Eqs. (1) and (2).

    • 3. Computation of the piecewise smooth inner surface of the optical element as it is described in the Section 2.2 and further fitting NURBS spline on it.

    • 4. Computation of the piecewise smooth outer surface of the optical element as it is described in the Section 2.3 and further fitting NURBS spline on it. After this step the complete geometry of the single optical element is obtained. The constructed single optical element generates the intensity distribution which is close to the required one. Accuracy of the required distribution generation depends on the number of points in the discrete light distribution given by Eqs. (1) and (2).

    • 5. Construction of the multilens containing a set of computed optical elements.

    3.1. Multi-lens array, generating intensity distribution with two symmetrical profiles

    The intensity pattern presented in Fig. 5 has two symmetrical cross-sectional profiles. The light distribution with such a directivity pattern corresponds to ME1 class of roadway lighting (or IESNA Type I accordingly), when the light poles are located in the middle of motorway and throw light in both driving directions (Fig. 6).


    Fig. 5 Q5.jpg

    Required intensity curves for road lighting with central pole arrangement.



    Fig. 6 Q6.jpg

    Scheme of road illumination with central pole arrangement.


    The required distribution was approximated by 900 points (𝑥𝑖,𝛷𝑖),𝑖=1,...,900(xi,Φi), i=1,...,900; afterwards, an optical element with refractive index 𝑛=1.493n=1.493 (polymethyl methacrylate) and 𝑘=0.5k=0.5 was designed. We used 𝑘=0.5k=0.5 because, in the case of using a point light source, such a value corresponds to the most efficient way to deflect rays, that has been proven in the [12]. The overall efficiency (the ratio of the flux passed the optical element to the source flux) was measured using a goniophotometer: it is equal to 89.6%. It should be noticed that the theoretical maximum in this case is 92% (Fresnel losses are at least 4% at each air-material interface, even in the case of normal ray incidence). A three-dimensional model of the computed element is shown in Fig. 7.


    Fig. 7Q7.jpg

     Single optical element model.


    Using the designed single optical element, a multi-lens array with 36 points was developed (Fig. 8) and manufactured (Fig. 9).


    Fig. 8 Q8.jpg

    Developed multi-lens array model.



    Fig. 9Q9.jpg

     Produced multi-lens array.


    Injection molding was used for producing the multilens. For the purpose of the best surface generation, the parameters of manufacturing process were optimized. These are as follows: melt temperature is 270 °С, mold temperature is 60 °С, melt injection velocity is 50 cm3/s, and injection boost time is 15 s. The manufacturing tolerance of the multi-lens array, produced with the parameters above, is 25 microns.

    To obtain the lighting distribution generated by produced lens, this optical element intensity distribution was measured in the photometry bench with Osram OSLON Square LED (GW CSSRM1.PC-MTNP-5H7I-1 type). The resulting intensity curves of the measured directivity pattern are shown in Fig. 10. The relative root-mean-square deviation of the generated by produced multilens (Fig. 9) light distribution from the required one is less than 6.6%.

    Fig. 10 Intensity curves generated by produced multilens.

    Download Full Size PPT Slide PDF

    The luminance and illuminance distributions on the roadway surface were simulated in DIALux software [17]. The luminaire configuration was as follows: the mounting height is 14 m, the overall number of lanes in each direction is two, and the width of each lane is 3.5 m.

    The requirements for this type of road and simulated values of average road surface luminance 𝐿𝑎𝑣Lav, overall luminance uniformity  , longitudinal luminance uniformity 𝑈𝐿UL and threshold increment 𝑇𝐼TI are presented in Table 1. The results obtained for the road surface type R3 in the DIALux software are shown for distances between luminaires of 35, 40, 45, and 50 m.

    Required and obtained values of road illumination criteria (example 1)


    Distance between luminaires, mLamp flux, lmAverage luminance, 𝐿𝑎𝑣Lav, cd/m2Overall luminance uniformity, 𝑈0U0Longitudinal luminance uniformity 𝑈𝐿ULThreshold increment 𝑇𝐼TI, %𝑆𝑅SR
    Experimental values
    35
    41000
    2.13
    0.45
    0.79
    5
    0.66
    40
    48000
    2.15
    0.44
    0.78
    6
    0.66
    45
    54000
    2.15
    0.44
    0.75
    7
    0.66
    50
    60000
    2.15
    0.42
    0.70
    8
    0.66
    Required values

    2.00.40.710>>0.5

    Table 1. Required and obtained values of road illumination criteria (example 1)


    The results presented in Table 1 confirm that the light distribution generated by the designed multi-lens array meets the requirements. This directivity pattern is universal and provides high performance with different configuration of the luminaires’ arrangement.

    3.2. Multi-lens array, generating intensity distribution with one symmetrical and one asymmetrical profile

    Let us now consider another example: an optical element generating light distribution on a six-lane road. As in the previous case, the lighting class is ME1 (or IESNA Type II accordingly). Figure 11 shows that this distribution has one asymmetrical profile in the transverse direction. The luminaires with such a directivity pattern are usually situated on the side of the road (Fig. 12).


    Fig. 11 Q11.jpg

    Required intensity curves for road lighting with side pole arrangement.



    Fig. 12 Q12.jpg

    Scheme of road illumination with side pole arrangement.


    The required intensity distribution was approximated by 700 points (𝑥𝑖,𝛷𝑖),𝑖=1,...,700(xi,Φi), i=1,...,700 and then an optical element with refractive index 𝑛=1.591n=1.591 (polycarbonate) and 𝑘=0.6k=0.6 was computed. The efficiency of the obtained solution is equal to 90.1%. A three-dimensional model of the designed element is presented in Fig. 13.


    Fig. 13 Q13).jpg

    Single optical element model.


    In Fig. 14, the three-dimensional model of the multilens, consisting of the optical elements depicted in Fig. 13, is shown. The produced sample is presented in Fig. 15.


    Fig. 14 Q14.jpg

    Developed multi-lens array model.



    Fig. 15Q15.jpg

     Produced multi-lens array.


    The following parameters of injection molding for producing the lens were used: melt temperature is 240 °С, mold temperature is 50 °С. Melt injection velocity and injection boost time are the same as in the case of PMMA: 50 cm3/s and 15 s respectively. The absolute tolerance of the produced multi-lens array is 25 microns also.

    Figure 16 demonstrates the intensity curves of lighting distribution, generated by the produced multilens (Fig. 15). The relative root-mean-square deviation of the generated light distribution from the required one is less than 7.8%. The luminance and illuminance distributions on the roadway surface were simulated in DIALux software. The luminaire configuration was as follows: the mounting height is 12 m, the boom length is 0.5 m and the elevation angle of the lamp is 0 °; the overall number of lanes in each direction is three, the width of each lane is 3.5 m.

    Fig. 16 Q16.jpg

    Intensity curves generated by produced multilens.


    The performance criteria and their measured values are presented in Table 2. The results obtained for the road surface type R3 in the DIALux software are shown for distances between luminaires of 30, 35, 40, and 43 m.

    Required and obtained values of road illumination criteria (example 2)


    Distance between luminaires, mLamp flux, lmAverage luminance, 𝐿𝑎𝑣Lav, cd/m2Overall luminance uniformity, 𝑈0U0Longitudinal luminance uniformity 𝑈𝐿ULThreshold increment 𝑇𝐼TI, %𝑆𝑅SR
    Experimental values
    30
    29000
    2.15
    0.63
    0.77
    6
    0.66
    35
    34000
    2.16
    0.58
    0.77
    7
    0.66
    40
    38000
    2.11
    0.54
    0.74
    8
    0.66
    43
    42000
    2.17
    0.54
    0.70
    8
    0.66
    Required values

    2.00.40.710>>0.5

    Table 2. Required and obtained values of road illumination criteria (example 2)


    We should note, that the intensity curves, obtained in the photometry bench, slightly differ from the required ones for both considered examples. This is caused by several aspects, such as an extended source, shrinkages appeared in molding process, the additional uncontrolled light flux reflected on the PCB surface. But as we can see from the comparison of Figs. 5, 10, 11, 16, the influence of these factors is not dramatical, because the difference between the required light distribution and measured ones has not exceeded 7.8% for both cases. Thus, we can say, that the presented modification of SQM method [12] enables the design of efficient smooth moldable optical elements for road lighting illumination.

    4. Conclusions

    We have demonstrated that the SQM modification [12] can be successfully applied to solution of such complicated and challenging problems as designing LED optical elements for street illumination. The presented approach enables computation of smooth and efficient optical elements with geometry, which is appropriate for further manufacturing by injection molding technology. As an example, two optical elements have been computed, molded and measured in the photometry bench. The analysis has shown that the obtained optical solutions provide high efficiency (more than 89% in both cases) and meet the highest illumination class (ME1) requirements for wide range of road configurations: in both cases the ratio of pole altitude to pole distance can vary from 2.5 up to 3.6. We highly recommend utilizing the proposed technique in commercial applications.

    Funding

    Russian Science Foundation project # 14-19-00969.

    References and links

    1. S. Zalewski, “Design of optical systems for LED road luminaires,” Appl. Opt. 54(2), 163–170 (2015). [CrossRef  [PubMed 

    2. M. A. Moiseev, L. L. Doskolovich, and N. L. Kazanskiy, “Design of high-efficient freeform LED lens for illumination of elongated rectangular regions,” Opt. Express 19(S3Suppl 3), A225–A233 (2011). [CrossRef  [PubMed 

    3. X. Hu and K. Qian, “Optimal design of optical system for LED road lighting with high illuminance and luminance uniformity,” Appl. Opt. 52(24), 5888–5893 (2013). [CrossRef [PubMed 

    4. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef  [PubMed 

    5. X.-H. Lee, I. Moreno, and C.-C. Sun, “High-performance LED street lighting using microlens arrays,” Opt. Express 21(9), 10612–10621 (2013). [CrossRef  [PubMed 

    6. P. Benitez, J. C. Minano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernandez, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004). [CrossRef 

    7. A. Bäuerle, A. Bruneton, R. Wester, J. Stollenwerk, and P. Loosen, “Algorithm for irradiance tailoring using multiple freeform optical surfaces,” Opt. Express 20(13), 14477–14485 (2012). [CrossRef  [PubMed 

    8. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013). [CrossRef  [PubMed 

    9. Z. Feng, L. Huang, G. Jin, and M. Gong, “Designing double freeform optical surfaces for controlling both irradiance and wavefront,” Opt. Express 21(23), 28693–28701 (2013). [CrossRef  [PubMed 

    10. R. Hu, Z. Gan, X. Luo, H. Zheng, and S. Liu, “Design of double freeform-surface lens for LED uniform illumination with minimum Fresnel losses,” Optik (Stuttg.) 124(19), 3895–3897 (2013). [CrossRef 

    11. M. A. Moiseev, S. V. Kravchenko, and L. L. Doskolovich, “Design of efficient LED optics with two free-form surfaces,” Opt. Express 22(S7Suppl 7), A1926–A1935 (2014). [CrossRef [PubMed 

    12. M. A. Moiseev, E. V. Byzov, S. V. Kravchenko, and L. L. Doskolovich, “Design of LED refractive optics with predetermined balance of ray deflection angles between inner and outer surfaces,” Opt. Express 23(19), A1140–A1148 (2015). [CrossRef  [PubMed 

    13. S. A. Kochengin and V. I. Oliker, “Computational algorithms for constructing reflectors,” Comput. Vis. Sci. 6(1), 15–21 (2003). [CrossRef 

    14. V. I. Oliker, Mathematical aspects of design of beam shaping surfaces in geometrical optics, Springer, 2003.

    15. V. Oliker, J. Rubinstein, and G. Wolansky, “Supporting quadric method in optical design of freeform lenses for illumination control of a collimated light,” Adv. Appl. Math. 62, 160–183 (2015). [CrossRef 

    16. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 2003).

    17. https://www.dial.de/en/software/