## Abstract

A systematic method is proposed for designing an optical system for road lighting using an LED and a freeform lens that is optimized to produce a certain luminance distribution on the road surface. The proposed design method takes account of the luminance characteristics of the road surface, the energy efficiency of the system, the glare problem of the luminaire and the effects of four adjacent luminaries illuminating a single road surface. Firstly, the road surface illuminance with a polynomial of cosine functions along the road is optimized to maximize *Q* (the ratio of the average luminance to the average illuminance) as well as satisfying the lighting requirements provided by CIE. Then, a smooth freeform lens with this optimized illuminance is designed based on the variable separation method and the feedback modification method. Results show that, from two typical observer positions on the 2-lane C2 class road, luminaires with these freeform lenses can provide *Q* values of 7.90 × 10^{−2} and 8.69 × 10^{−2}, the overall road surface luminance uniformity of 0.55 and 0.56, the longitudinal road surface luminance uniformity of 0.72 and 0.79, and the glare factors of 10.06% and 6.73% .

© 2010 OSA

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### Equations (22)

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(1)
$$L(\beta ,\gamma )={\displaystyle \sum _{k=1}^{K}\frac{r(\beta ,\gamma )}{{10}^{4}\cdot {\mathrm{cos}}^{3}\gamma}\cdot {E}_{k}(c,\gamma )}$$
(2)
$$L({x}_{i},{y}_{j})={\displaystyle \sum _{k=1}^{K}\frac{{r}_{k}({x}_{i},{y}_{j})}{{10}^{4}\cdot {\mathrm{cos}}^{3}{\gamma}_{k}({x}_{i},{y}_{j})}\cdot {E}_{k}({x}_{i},{y}_{j})}$$
(3)
$${E}_{av}=\frac{{\displaystyle \sum _{i=1}^{N}{\displaystyle \sum _{j=1}^{M}E({x}_{i},{y}_{j})}}}{NM},E({x}_{i},{y}_{j})={\displaystyle \sum _{k=1}^{\text{K}}{E}_{k}({x}_{i},{y}_{j})}$$
(4)
$${L}_{av}=\frac{{\displaystyle \sum _{i=1}^{N}{\displaystyle \sum _{j=1}^{M}L({x}_{i},{y}_{j})}}}{NM}$$
(5)
$${E}_{0}=\frac{\mathrm{min}(E({x}_{i},{y}_{j}))}{{E}_{av}},i=1,2,\mathrm{...}N,j=1,2,3,\mathrm{...}M$$
(6)
$${U}_{0}=\frac{\mathrm{min}(L({x}_{i},{y}_{j}))}{{L}_{av}},i=1,2,\mathrm{...}N,j=1,2,3,\mathrm{...}M$$
(7)
$${U}_{L}=\frac{\mathrm{min}(L({x}_{i},{y}_{center}))}{\mathrm{max}(L({x}_{i},{y}_{center}))},i=1,2,3,\mathrm{...}N$$
(8)
$${L}_{v}=10\cdot {\displaystyle \sum _{k=1}^{K}\frac{{E}_{ek}}{{\theta}_{ek}{}^{2}}}$$
(9)
$$TI=65\frac{{L}_{v}}{{L}_{av}^{0.8}}$$
(10)
$$\begin{array}{c}{E}_{0}(x,y)={E}_{x}(x)\cdot {E}_{y}(y)\\ =\left({\displaystyle \sum _{i=1}^{J}{a}_{i}{\mathrm{cos}}^{{n}_{i}}(\frac{\pi x}{2{x}_{\mathrm{max}}}})\right)\cdot 1\\ ={\displaystyle \sum _{i=1}^{J}{a}_{i}{\mathrm{cos}}^{{n}_{i}}(\frac{\pi x}{2{x}_{\mathrm{max}}}})\end{array}$$
(11)
$$\mathrm{max}(Q({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J}))=\mathrm{max}(\frac{{L}_{av}({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})}{{E}_{av}({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})})$$
(12)
$${L}_{av}({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})\ge {L}_{av}{}_{T}$$
(13)
$${U}_{0}({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})\ge {U}_{0}{}_{T}$$
(14)
$${U}_{L}({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})\ge {U}_{LT}$$
(15)
$$TI({a}_{1},{a}_{2},\mathrm{...},{a}_{J},{n}_{1},{n}_{2},\mathrm{...}{n}_{J})\le T{I}_{T}$$
(16)
$$\underset{\Omega}{\iint}{I}_{0}{\mathrm{cos}}^{2}u\mathrm{cos}v}dudv={\displaystyle \underset{D}{\iint}{E}_{0}(x,y)}dxdy$$
(17)
$${u}_{j+\text{1}}=f({\displaystyle \sum _{i=1}^{n}{\displaystyle \sum _{j=1}^{j}{E}_{0}({x}_{i},{y}_{j})}})$$
(18)
$${v}_{i+1}=\text{g}({\displaystyle \sum _{j=1}^{m}{\displaystyle \sum _{i=1}^{i}{E}_{0}({x}_{i},{y}_{j})}})$$
(19)
$${\eta}_{l}({x}_{i},{y}_{j})={\beta}_{l}({x}_{i},{y}_{j})\cdot {\beta}_{l-1}({x}_{i},{y}_{j})\cdot \cdot \cdot {\beta}_{1}({x}_{i},{y}_{j})$$
(20)
$${\beta}_{l}({x}_{i},{y}_{j})=\frac{{E}_{0}({x}_{i},{y}_{j})}{(1-\lambda ){E}_{0}({x}_{i},{y}_{j})+\lambda E{\text{'}}_{l}({x}_{i},{y}_{j})}$$
(21)
$${E}_{l+1}({x}_{i},{y}_{j})={\eta}_{l}({x}_{i},{y}_{j})\cdot {E}_{0}({x}_{i},{y}_{j})$$
(22)
$$Flu{x}_{waste}=({E}_{avopti}-{E}_{avsimu})\times W\times S|{}_{{L}_{av}=1.5cd/{m}^{2}}$$