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Mechanical Engineering Department, Kun Shan University, Taiwan

- *Corresponding Author:
- Ku Chin Lin

Mechanical Engineering Department

Kun Shan University, Taiwan

**Tel:**886-6-2057122

**E-mail:**[email protected]

**Received date:** January 13, 2015; **Accepted date:** February 02, 2015; **Published date:** February 05, 2015

**Citation:** Lin KC (2015) Freeform Lens Design for Illumination with Different Luminance Intensities. J Comput Sci Syst Biol 8:099-103. doi: 10.4172/jcsb.1000175

**Copyright:** © 2015 Lin KC. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

**Visit for more related articles at** Journal of Computer Science & Systems Biology

Mapping of equi-luminous fluxes from a point source through a freeform lens for illumination with different luminance intensities is studied. The freeform surface is associated with the mapping and the grids to design on a target plane. Relocation of the target grids designed for uniform illumination is proposed by comparing the grids with reference and desired ones for interpolation to achieve the desired illumination distribution. Target-grids relocation for rectangular illumination with vertical, horizontal, tilt, circular, ring and rectangular cut-off for two different luminance intensities is demonstrated. The target grids after relocation are smooth in distribution, and the lens with a smooth freeform surface is achieved. The adequacy of the proposed methodology is proved by simulation.

Luminance intensities; Target-grids

A freeform lens has variety of applications such as LED TV, microprojector, auto fog light and headlight, etc. [1,2]. Methods for freeform lens design are primarily based on the grids mapped by equi-luminous flux rays on a target plane. Typical methods may include those to calculate global surface grids simultaneously [3] and local surface grids successively [4,5]. The surface grids are required to be smooth to ensure the applicability of the edge-ray principle [6,7] and the satisfaction of the integrability condition [8].

Freeform lens design for uniform illumination has been widely studied, but for illumination with different luminance intensities the studies are few. Lenses with discrete freeform surfaces may be designed to generate illumination with different luminance intensities [4] However, it could be inadequate in practice if the size of the source is a significant factor, and the manufacturing cost is still high currently [9,10].

A pioneer work to design a freeform lens for illumination with two different luminance intensities was proposed [11]. Three alphabets “oec” with higher luminance intensity than the background is achieved using a tailored-made lens. However, an efficient approach to construct the grids mapped by equi-luminous flux rays was missing. It deserves an advanced study to make the design of a lens for illumination with two different luminance intensities become more efficient. Henceforth, initiation of such a study is proposed here.

In this study, a new approach to design the grids mapped by equiluminous flux rays is proposed. The grids to design are called the target grids (TG). Illumination grid (IG) is introduced over the illumination area. The IG is associated with the luminance intensity distribution. For example, it is uniformly distributed for uniform distribution of luminance intensity. Uniformly-distributed IG will be defined as reference illumination grids (RIG). The IG associated with desired luminance distribution with different intensities is named desired illumination grid (DIG). The proposed approach to calculate the TG for illumination with two different intensities is based on relocation of the TG for uniform illumination and interpolation of the DIG among the RIG.

Typical candela distribution of an LED light is of a Lambertian type. The distribution and its accumulation after normalization are given in **Figure 1**. The accumulation is equally divided into N_{?} divisions along the inclination and N_{θ} divisions along the azimuthal,

(1a)

(1b)

where *φ _{s}* denotes the inclination angle of a light ray; e the candela distribution;

**Figure 2:** Target grids for design of freeform lenses for different contours of
uniform illumination. a. Target grids for circular contour of uniform illumination.
b. Target grids for rectangular contour of uniform illumination. c. Target grids
for triangular contour of uniform illumination. d. Target grids for trapezoidal
contour of uniform illumination.

**Figure 3:** Freeform lenses designed based on the target grids shown in Figure
2 for different contours of uniform illumination. a. Lens for circular contour of
uniform illumination. b. Lens for rectangular contour of uniform illumination.
c. Lens for triangular contour of uniform illumination. d. Lens for trapezoidal
contour of uniform illumination.

**Figure 4:** Simulated luminance intensity distribution generated by the lens
shown in Figure 3 for different contours of uniform illumination. a. Luminance
intensity distribution generated by the circular lens. b. Luminance intensity
distribution generated by the rectangular lens. c. Luminance intensity
distribution generated by the triangular lens. d. Luminance intensity distribution
generated by the trapezoidal lens.

An example of RIG for rectangular uniform illumination is in **Figure 5a**. Then RIG is uniformly-s-distributed grids and described below:

(2a)

(2b)

**Figure 5:** Reference and desired illumination grids for rectangular illumination
with vertical, horizontal and tilt cut- off. a. Reference illumination grids
for rectangular illumination. b. Desired illumination grids for rectangular
illumination with vertical cut-off. c. Desired illumination grids for rectangular
illumination with horizontal cut-off. d. Desired illumination grids for rectangular
illumination with tilt cut-off.

where W denotes the width of the rectangle; H the height of the rectangle R_{x}(k) and R_{y}(k); the distance of k^{th} grid from the left edge and the bottom edge of the rectangle, respectively; N_{x} the number of grids distributed along the x axis; N_{y} the number of grids along the y axis. The DIG for illumination with a vertical cut-off line at x=0 is shown in **Figure 5b.** The DIG for illumination with a horizontal cut-off line at y=0 is in Figure 5c. The DIG for illumination with a tilt cut-off line at (x,y)=(0,0) is shown in Figure 5d where the slope of the line is 2. The TG for rectangular uniform illumination is in **Figure 6a**. The distribution density at the right is twice higher than that at the left and it is described as below:

(3)

**Figure 6:** Target grids for rectangular illumination with vertical, horizontal and
tilt cut-off. a. Target grids for rectangular contour of uniform illumination.
b. Target grids for rectangular illumination with vertical cut-off. c. Target
grids for rectangular illumination with horizontal cut-off. d. Target grids for
rectangular illumination with tilt cut-off.

where D_{x}(k) the distance of the kth grid from the left edge of the rectangle.

The distance of each of the TG from the left edge of the rectangle is given below:

(4)

where T_{x}(i,j) denotes the x coordinate of the (i, j)^{th} grid of the TG.

By comparing the x coordinate of each of the TG with R_{x}(k), from k=1 until R_{x}(k*)< T_{d}(i,j) ≤ R_{x}(k*+1), and we have

(5a)

(5b)

Where

(5c)

and T_{y}(i, j) denotes the y coordinate of the (i,j)^{th} grid of the TG; (T_{x}',T_{y}') denotes the TG after interpolation and it is in **Figure 6b**. The DIG for illumination with a horizontal cut-off line at y=0 is in Figure 5c. The distribution density at the bottom is twice higher than that at the top, and it is described as below:

(6)

where D_{y}(k) the distance of the k^{th} grid from the bottom edge of the rectangle.

The distance of the TG from the bottom edge of the rectangle is given below:

(7)

where T_{y}(i, j) denotes the y coordinate of the (i, j)^{th} grid of the TG.

By comparing the y coordinate of each of the TG with R_{y}(k), from k=1 until R_{y}(k*)<T_{d}(i,j) ≤ R_{y}(k*+1), and we have

(8a)

(8b)

Where

(8c)

and T_{x}(i, j) denotes the x coordinate of the (i, j)^{th} grid of TG; the TG after interpolation is shown in Figure 6c.

The TG after interpolation is shown in Figure 6d. The DIG for illumination with a tilt cut-off line at (x,y)=(0,0) is shown in Figure 5d where the slope of the line is 2. The designed freeform lenses for the above DIG are shown in **Figure 7**. The outer contours of the lenses agree with the unbalanced illumination. Simulation results of 50,000 tracing rays using Light Tools for N_{?}=50 and N_{θ}=200 are shown in **Figure 8**, and the desired illumination with two different luminance intensity is achieved. For illumination with a circular, ring and rectangular cutoff area, the TG after relocation is shown in **Figure 9**. The designed freeform lenses are in **Figure 10**. The freeform surfaces of the lenses are popped-up with a smooth circular, ring and rectangular area. Simulation results of luminance intensity distribution generated by the lenses are in **Figure 11**, and illumination with desired distribution of luminance intensity is achieved.

**Figure 7:** Freeform lenses designed based on the target grids shown in Figure
6 for rectangular, illumination with vertical, horizontal and tilt cut-off. a. Lens for
rectangular illumination with vertical cut-off. b. Lens for rectangular illumination
with horizontal cut-off. c. Lens for rectangularillumination with tilt cut-off.

**Figure 8:** Simulated luminance intensity distributions generated by the lenses
shown in Figure 7 for rectangular illumination with vertical, horizontal and tilt
cut-off. a. Luminance intensity distribution with vertical cut-off. b. Luminance
intensity distribution with horizontal cut-off. c. Luminance intensity distribution
with tilt cut-off.

**Figure 9:** Target grids for rectangular illumination with circular, ring
and rectangular cut-off contours. a. Target grids for rectangular
illuminationrectangular with circular cut-off. b. Target grids for rectangular
illumination with ring cut-off. c. Target grids for rectangular illumination with
rectangular cut-off.

**Figure 10:** Freeform lenses designed based on the target grids shown in
Figure 9 for rectangular illumination with circular, ring and rectangular cut-off.
a. Lens for rectangular illumination with circular cut-off. b. Lens for rectangular
illumination with ring cut-off. c. Lens for rectangular illumination with
rectangular cut-off.

**Figure 11:** Simulated luminance intensity distribution generated by the
lenses shown inFigure 10 for rectangular illumination with circular, ring and
rectangular cut-off. a. Luminance intensity distribution with circular cut-off.
b. Luminance intensity distribution with ring cut-off. c. Luminance intensity
distribution with rectangular cut-off.

Freeform lens design for illumination with different luminance intensities is studied. The proposed approach is an extension of the previous one for uniform illumination. Examples of the TG of equiluminous fluxes for uniform illumination are given in Figure 2. The IG distribution is associated with the distribution of luminance intensity. Examples of RIG and DIG are given in Figure 5. Methods to relocate the TG for illumination with different luminance intensities are proposed by relocating the TG for uniform illumination and interpolation of the DIG among the RIG. The TG for rectangular illumination with vertical, horizontal, tilt, circular, ring and rectangular cut-off has been demonstrated in Figures 6 and 9. The *φ ^{*}_{p}* and

This work was supported by the National Science Council, Taiwan, R.O.C. under the grant NSC-103-2221-E-168-017-.

- Cvetkovic A, Oliver D, Julio C, Pablo B, Juan CM
*et al*. (2006)Etendue-preserving mixing and projection optics for high-luminance LEDs, applied to automotive headlamps.Opt Express 14:113014-113020. - Chen F, K Wang, Qin Z, Wu D, Luo X
*et al.*(2010) Design method of high-efficient LED headlamp lens.Opt Express 18: 20926-20938. - Ding Y, Liu X, Zheng ZR, GuPF (2008)Freeform LED lens for uniform illumination.OptExpress 16: 12958-12966.
- Wang K, Liu S, Chen F, Liu ZY, Luo XB (2009) Effect on manufacturing defects on optical performance of discontinuous freeform lens.Opt Express 17: 5457-5465.
- Wang K, Chen F, Liu ZY, Luo XB, Liu S (2010)Design of compact freeform lens for application specific lighting-emitting diode packaging. Opt Express 18: 413-425.
- Davies PA (1994)Edge-ray principle of nonimaging optics. JOptSocAm A11: 1256-1259.
- Ries H,RablA (1994) Edge-ray principle of nonimaging optics. J Opt SocAm A 11: 2627-2632.
- FrankotR, ChellappaR (1988)A method for enforcing integrability in shape from shading algorithms. IEEE Trans Pattern Anal Mach Intell 10: 439-451.
- Fang FZ, Zhang XD, Hu XT (2008) Cylindrical coordinate machining of optical freeform surfaces. Opt Express 16: 7323-7329.
- Xia L, Yin S, Dong X, Du C (2010)Multi-segment freeform LED uniform lens with low reflective loss.Proc of SPIE 7849: 784910-784916.
- Ries H, Muschaweck J (2002) Tailored freeform optical surfaces. J Opt Soc Am A 19: 590-595.
- Lin KC (2013)Smooth Energy Mappings of Freeform Lens Design for Non-Circular Distribution of LuminanceUKSim-AMSS 15th International Conference on Modeling and Simulation (UKSIM 2013) Cambridge University UK 10-12 April. 150-154.
- Lin KC(2014)Smooth Energy Mappings for Freeform Lens Design. Optical Review21: 639-641.

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