## Abstract

The illumination pattern of an LED street light is required to have a rectangular distribution at a divergence-angle ratio of 7:3 for economical illumination. Hence, research supplying a secondary optics with two cylindrical lenses was different from free-form curvature for rectangular illumination. The analytical solution for curvatures with different ratio rectangles solved this detail by light tracing and boundary conditions. Similarities between the experiments and the simulation for a single LED and a 9-LED module were analyzed by Normalized Cross Correlation (NCC), and the error rate was studied by the Root Mean Square (RMS). The tolerance of position must be kept under ± 0.2 mm in the x, y and z directions to ensure that the relative illumination is over 99%.

© 2013 OSA

## 1. Introduction

The characteristics of light-emitting diodes (LEDs) that make them an energy-efficient and long-lasting light source for general illumination have attracted a great deal of attention from the lighting industry [1–3]. As everyone knows, LEDs have the advantages of environmental protection, long lifetime, fast response time (μs) and good mechanical properties [4–6]. Their high luminance and the wide span of the dominant wavelengths within the entire visible spectrum mean that people have high expectations for LEDs. People have found a wide range of applications for LEDs in Solid State Lighting (SSL), like: street lighting, back lighting, sign lighting, interior lighting, and so on [7–10]. Although LEDs are already widely used for street lighting, some problems still need to be solved related to the illumination pattern, luminous efficiency, lighting uniformity, etc. LEDs could not be directly used for street lighting due to the Lambertian radiation distribution and non-uniform illumination; therefore, a secondary optics is necessary for designing a rectangular radiation pattern [11].

In general, a first optical design means the design-on-board and a second optical design is the design-on-module. Many researchers have used aspheric or freeform surfaces to design a second optical lens, thereby realizing the illumination pattern, luminous efficiency and uniform rectangular shape for street lighting [12–14], but the fabrication of the aspheric or freeform surfaces is very difficult and expensive. Construction of a freeform lens is complex because it is based on a curve to generate other curves [15]. It is important to consider whether the design can be manufactured for a freeform lens, and there is no principle to suggest which curve is valid. Hence, the research focused on designing a suitable spherical surface for the LED secondary optics. In this paper only one simple method was employed using two cylindrical lenses on the secondary optics to create a rectangular and uniform illumination. The method also only modified the curvature radius to get a different x-y angle ratio for the rectangular illumination. A cylindrical lens and the oblique incident principle were successfully used to obtain the secondary optics design for rectangular illumination. Results from the Normalized Cross Correlation (NCC) between the experiments and the simulation for the single LED and 9-LED module showed them to be similar in extent, reaching 98.46% and 97.30%, and the error rate could be reduced to 4.36% and 4.87%, respectively.

## 2. Lens construction and design flow

#### 2.1. Lens construction

A spherical lens is the most convenient design and is usually used for street light secondary optics. The illumination pattern is always a circle, with the focus at the center of the illumination distribution. The illumination is not usually uniform and wastes lighting by shining outside the desired area on the street. Therefore, an economical illumination pattern for LED street lights is a rectangular shape, as shown in Figs. 1(a)
-1(b). An aspheric and freeform surface is usually used for the design of rectangular illumination, but the secondary optics is complex and expensive. Hence, a design with two cylindrical spherical lenses was used to ensure uniform illumination distribution. Table 1
shows a comparison list between freeform and cylindrical lenses used for rectangular illumination. The light source was a 1 W white LED (Cree XLamp XP-G LED), and the mockup samples for both the single LED and 9-LED module are shown in Figs. 2(a)
-2(b). A cylindrical spherical lens was designed using 3-D drawing software (SolidWork), shown in Fig. 3
. In Fig. 4(a)
it can be seen that the output light slants to the outside due to the cylindrical lens. Figure 4(b) shows the divergence angles of *θ _{x}* and

*θ*of the rays passing through the secondary optics lens.

_{y}*θ*and

_{x}*θ*are equal to the divergence angles of

_{y}*U*and

*V*, respectively, for rectangular distribution in

*x*and

*y*. Therefore, the output lighting can be focused on the outside regions to meet the requirement for a rectangular-shaped illumination pattern with a divergence-angle ratio of 7:3 in the

*x*and

*y*directions. The output lighting in the outside area has increased efficiency due to the oblique incident lighting for illumination uniformity.

#### 2.2. Design flow

An oblique marginal ray was considered for the light tracing. Figures 5(a)
and 5(b) show the light tracks on the first and second surface, respectively, in the *x* direction (horizontal axis). The cylinder lens was assumed to be a spherical surface of radius *r* centered at a point *C* and separating the media of refractive index n. In Fig. 5(a), it was assumed that *Q* was the point on the first surface for a marginal incident ray, and the ray in the meridional plane was specified by the field angle *U* made with the axis, and by the distance *M* = *OB* between pole *O* and point *B* where it met the axis. *I* would be the angle between the incident ray and the normal *QC*. The distances of *r*, *M*, *M’* and d were taken to be positive when points *C*, *B*, *B’* and *B”* were to the right of point *O*, and the light was assumed to be incident from the left. The angles of *U*, *U’* and *U”* were considered to be positive if the rays of *QB*, *QB’ EU’* and *EU”* could be brought counterclockwise into coincidence with the horizontal axis at points *B*, *B’*, *L* and *S*, respectively. Angles *I* and *I’* were taken to be positive if the incident and the refracted rays could be made to coincide with the normal line *QC* by a counterclockwise rotation about point *Q*. These definitions are the same in the y direction, as shown in Figs. 6(a)
and 6(b). From triangle *QB’C* in Fig. 5(a), angle *U* could be found by the formula for an exterior angle:

*YE*was the tangent line for the second surface at point

*E,*and angle

*θ*was the intersection of line

*YE*and the center axis. Angle

*U”*was the divergence angle of the LED and the relationship between

*U’*and

*U”*was derived by Snell’s Law:

*U’*could be expressed as:

*I*and

*I’*could be found by Snell’s Law:and substituting

*I’*and

*U’*into Eq. (1) such that:

*U*was given as 70 degrees and angle

*U”*was decided by the spec. of the LED. In order to obtain angle

*I*,

*I*was calculated by a numerical solution. When angle

*I*had a solution, it meant that the maximum divergent ray was in existence. The next step was to find the relationship between

*R*and

_{x}*U”*. Triangles

*QGC*and

*B’GC*had the same line

*GC,*as seen in Fig. 5(a) and the relationship could be written as follows:From Fig. 5(b) one can see that triangles

*EYH*and

*ESH*also had the same length

*EH*and gave:Line

*B’K*could be written by the Pythagorean Theorem in triangle

*B’KL*. Line

*B’K*was considered related to the spherical lens:Since triangles

*EHL*and

*EHS*had the same line

*EH*, the relationship between points

*F’*and

*F”*could be written as:

*SY*as

*T*and substituted

*F”*by Eq. (10):

*R*could be found by substituting

_{x}*I’*,

*U’*and

*U”*into Eq. (11):

*V’*could be obtained using the formula for the exterior angle of triangle

*ADD’*in Fig. 6(a):

*V”*was the divergent angle of the LED in Fig. 6(b). Snell’s Law was required to find this marginal ray passing through the second surface:Angle

*J’*could be found by Snell’s Law on the first surface:The equation could be written by substituting angles

*J’*and

*V’:*

*V*was the divergent angle of the marginal ray passing through the lens. Then, distance

*N’*could be assembled by angles

*J’*and

*V’*:Triangles

*HWS*and

*HWD’*had the same line HW, as seen in Fig. 6(b). Thus, the distance

*T*was obtained by the following:Distance

*N”*could be expressed as:Distance

*N’*was the sum of lines

*OW*and

*WS,*as seen in Fig. 6(b). Thus,

*R*could be related as in the following equation:Curvature

_{y}*R*was obtained by substituting distance

_{y}*N’*in Eq. (16) and distance

*N”*in Eq. (18):

A high power 1 W LED was selected as the light source for the street lamp. The design and simulation flow are exhibited in Fig. 7
. First, the size of the secondary optics was determined by the LED size and boundary conditions. Then, the material used was optical level polycarbonates (PC) (143R-111, GE Corp.) with a refractive index of 1.586. Second, the divergent angles of both *U* and *V* in x and y directions were determined, respectively, arriving at the curvatures of *R _{x}* and

*R*

_{y}_{.}The divergence angle of

*U*is an off-axis system regarding the light source S in Fig. 5. Effective parameters included the distance

*d*between the LED and secondary optics and angle θ between the intersection of line

*YE*and the center axis in Fig. 5. The divergence angle of

*V*is an on-axis system for the light source S in Fig. 6. The effective parameter was only the distance

*d*between the LED and the secondary optics. Third, the curvatures in the connected and boundary regions were optimized. Ray Tracing Software Trace-Pro was used to analyze the stray lighting. Finally, the uniformity ratio must be lower than 1.3 in order to finish the optimal process. The dimension size of the secondary optics was 15x13x4 mm.

## 3. Result and discussion

#### 3.1. Discussion of curvatures and divergence angles

According to Eqs. (11) and (20), Fig. 8(a)
shows the relationship between curvature *R _{x}* and divergence angle

*U*with different distances

*T*ranging from 1 to 5 mm. The tangent angle

*θ*was fixed at 25 degrees and distance

*d*was kept at 5 mm. Curvature

*R*was proportional to divergence angle

_{x}*U,*with the distance

*T*being over 3 mm. The slope variation was rapidly increased with the increase of curvature

*R*, at a distance

_{x}*T*of over 3 mm. If the slope was large, the efficiency solutions would become small for curvature

*R*and divergence angle

_{x}*U*. This phenomenon was due to the high value of distance

*T*between the lens and the light source which limited the divergent angle

*U*at a large curvature

*R*. Therefore, the best design for curvature

_{x}*R*at distance

_{x}*T*would be under 3 mm because divergence angle

*U*could vary from 68 to 90 degrees. Figure 8(b) shows the relationship between curvature

*R*and divergence angle

_{x}*U,*with the size of tangent angle

*θ*varying from 21 to 25 degrees. Distance

*T*was fixed at 1 mm and distance

*d*was fixed at 5 mm. It could be observed that curvature

*R*and divergence angle

_{x}*U*decreased with the increase of tangent angle

*θ*varying from 21 to 25 degrees. If curvature

*R*was fixed as large as 8.5 mm, the small tangent angle would have a large divergence angle of 70 degrees. Figure 9 shows the relationship between curvature

_{x}*R*and divergence angle

_{y}*V,*with different distances

*T*from 1 to 5 mm. The distance d was fixed at 1 mm. This was the same as what was seen in Fig. 8(a), where there was a high value for distance

*T*between the lens and light source limiting the divergent angle

*V*at the large curvature

*R*. If curvature

_{y}*R*was 10 mm and

_{y}*T*was 2 mm, the large divergence would have an angle of 30 degrees.

#### 3.2. Quantification of similarity between simulation and results

Figures 10(a) -10(b) exhibit the simulation and experimental results for street lighting illumination patterns for a single LED and Figs. 11(a)-(b) show them for a 9-LED module. The illumination patterns simulated by optical software of Trace-Pro with a million rays and the experimental measurement were done using a Goniophotometer (LSI Model:7500). The experimental results were very close to the simulation results for both the single LED and 9-LED module. In order to determine the extent of similarity we introduced a correlation expression for the Normalized Cross Correlation (NCC) [18,19] to our measurement results. The NCC results clearly exhibited a similarity between the experimental and simulated data by a probability, i.e.:

*X*and

_{mn}*Y*are the values of the experimental and simulated data; and $\overline{X}$and $\overline{Y}$are the mean values of the experimental and simulated data across the angular range. The NCC could reach 98.46% and 97.30% for the single LED and 9-LED module, respectively. The error rate can be obtained using the Root Mean Square (RMS) Method for experimental and simulation results. The RMS error between the experimental and simulation results can be calculated on a range of S points over the domain. It can be expressed by:

_{mn}*S*indicates the points calculated on the range. We also found that the extent of error could be reduced to 4.36% and 4.87% by the RMS error mode for the single LED and 9-LED module, respectively. The RMS made it obvious that both error rates were under 5% indicating that the simulation is approximate to the experiment. Figures 12(a) -12(b) and 13(a) -13(b) indicated the extent of similarity in both the horizontal and vertical directions for the single LED and 9-LED module, respectively. The experimental and simulated data were also very close to each other. The uniformity according to the IESNA [20] was:Symbols,

*E*and

_{min}*E*, indicate the minimum and average luminosity of these points in the x and y directions, respectively. In Figs. 13(a) and 13(b) the uniformity calculated by the experimental data was shown as 1.28 which was lower than 3 suggested by the IESNA. Then, in Figs. 14(a) and 14(b) the uniformity calculated by the experimental data was shown as 1.16. The average uniformity could be lower than 1.3 for the single LED and 9-LED module.

_{av.}#### 3.3. Illumination pattern results

As is commonly known, a spherical lens focuses lighting into a central point, but a cylindrical lens focuses lighting into a line. In other words, we would see different illumination patterns for the spherical and cylindrical lenses, as shown in Figs. 14(a)-14(b). The illumination pattern of the spherical lens produced a Lambertian distribution, while the cylindrical lens produced a bat-wing distribution. In the design of a cylindrical lens, the illumination could extend to 70 degrees and 30 degrees in the x and y directions separately, but there was uniform illumination for the rectangular distribution.

#### 3.4. Optical efficiency and tolerance analysis

The optical efficiency of the single LED could be reduced from 95.37 lm/W (without a secondary optics) to 88.68 lm/W (with a secondary optics) by an integration spherical meter. However, we were also interested in the efficiency of the fabrication error. Since the tolerance analyses of positions in the *x, y* and *z* directions were carried out for a 1 W single LED, Fig. 15
shows that the relative illumination at the position in the *z* direction was more sensitive than in the *x* and *y* directions. For a maximum distance error of 1 mm, we found the variation in the relative illumination to follow the order of *z>y>x*. We believe that the reason for this was that the distance *z* could control the focus length between the cylindrical lenses and the LED lighting source, and the distance of y could be affected by the cylindrical curvature of the LED light source. Hence, the distance direction of *z* and y had more sensibility to *x* at the relative illumination. In order to ensure that the relative illumination is over 99%, the distance errors of *x*, *y* and *z* must be kept under ± 0.2 mm.

## 4. Conclusions

This study investigated a secondary optics design with two cylindrical spherical lenses on the illumination pattern and uniformity for an LED street light. First, a rectangular illumination pattern, with a divergence-angle ratio of 7:3 in the x and y directions and a uniformity of illumination, was required. Hence, a secondary optics design with two cylindrical lenses was used to obtain the desired lighting distribution. Then, an optimal simulation of the curvatures was carried out to achieve a rectangular illumination with a bat-wing shaped pattern. Second, it was found that the extent of similarity between the experimental and simulation results could reach 98.46% and 97.30% using NCC. The error rate could be reduced to 4.36% and 4.87% by RMS for the single LED and 9-LED modules, respectively. The average uniformity could be lowered to 1.28 and 1.16 for the single LED and 9-LED module, respectively. Finally, the distance tolerance of the z direction was more sensitive than the x and y directions for relative illumination in the tolerance analyses. The tolerances of x, y and z must be kept under ± 0.2 mm for the relative illumination to be over 99%.

## Acknowledgments

The authors would like to thank the National Science Council of the Republic of China, Taiwan for financially supporting this research under Contract No. NSC101-2220-E-224-002 and NSC 101-2220-E-224-001-.

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