## Abstract

We developed an analytical method of illuminance formation for mixed-color LEDs in a rectangular light pipe in order to derive American National Standards Institute (ANSI) light uniformity, ANSI color uniformity, and color difference of light output using photometry, nonimaging, and colorimetry. The analytical results indicate that the distributions of illuminance and color difference vary with different geometric structures of light pipes and the location of the light sources. It was found that both the ANSI light and the ANSI color uniformity on the exit plane of the light pipe are reduced exponentially with the increase in length of the light pipe.
It is evident that a length scale *L*∕*A* greater than unity assures that the mixed-color LED sources on the entrance plane are uniformly illuminated with acceptable uniform brightness and color on the exit plane of the rectangular light pipe, where *L* is the length of the light pipe, and *A* is a constant, which is a geometric parameter for the scale unit of the light pipe's input face. Furthermore, the ANSI light uniformity can be minimized, and the ANSI color uniformity can be maximized under the condition of multilight-source locations
$P=Q=\pm A/4$, where *P* and *Q* are the coordinates along the long axis and the short axis, respectively, with one being the entrance plane of the light pipe. We can conclude that the optimum form factor of the light pipe is a square shaped cross section, with the length scale *L*∕*A* being equal to unity and with multilight sources located individually on positions of *A*∕4 in order to achieve very uniform illuminations with the highest light efficiency and compact package for the optical system with mixed-color LEDs, where *L* is the length of the light pipe.

© 2008 Optical Society of America

Full Article |

PDF Article
### Equations (33)

Equations on this page are rendered with MathJax. Learn more.

(1)
$${J}_{\mathrm{\Omega}}=J\left(\theta \right)={J}_{0}\text{\hspace{0.17em} cos \hspace{0.17em}}\theta \text{, \hspace{1em}}-90\xb0\le \theta \le 90\xb0\text{,}$$
(2)
$$\text{cos \hspace{0.17em}}\theta =\frac{L}{\sqrt{{L}^{2}+\left({x}^{2}+{y}^{2}\right)}}=\frac{L}{R}\text{.}$$
(3)
$$H\left(x,y\right)=J\left(\theta \right)\frac{{\text{cos}}^{3}\text{\hspace{0.17em}}\theta}{{L}^{2}}={J}_{0}\times \frac{{L}^{2}}{{\left({L}^{2}+{x}^{2}+{y}^{2}\right)}^{2}}\text{.}$$
(4)
$${H}_{00}\left(x,y\right)={J}_{0}\times \frac{{L}^{2}}{{\left({L}^{2}+{x}^{2}+{y}^{2}\right)}^{2}}\text{, \hspace{1em}}-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{.}$$
(5)
$${H}_{10}^{\prime}\left(x,y\right)={J}_{0}\times \frac{\rho {L}^{2}}{{\left({L}^{2}+{x}^{2}+{y}^{2}\right)}^{2}}\text{, \hspace{1em}}\frac{a}{2}\le x\le \frac{3a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{,}$$
(6)
$${H}_{20}^{\prime}\left(x,y\right)={J}_{0}\times \frac{{\rho}^{2}{L}^{2}}{{\left[{L}^{2}+{x}^{2}+{y}^{2}\right]}^{2}}\text{, \hspace{1em}}\frac{3a}{2}\le x\le \frac{5a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{,}$$
(7)
$${H}_{10}\left(x,y\right)={J}_{0}\times \frac{\rho {L}^{2}}{{\left[{L}^{2}+{\left(-x+a\right)}^{2}+{y}^{2}\right]}^{2}}\text{,}-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{,}$$
(8)
$${H}_{20}\left(x,y\right)={J}_{0}\times \frac{{\rho}^{2}{L}^{2}}{{\left[{L}^{2}+{\left(x+2a\right)}^{2}+{y}^{2}\right]}^{2}}\text{,}$$
(9)
$$-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{.}$$
(10)
$${H}_{\pm 1\pm 1}\left(x,y\right)={J}_{0}\times \frac{\rho \rho {L}^{2}}{{\left[{L}^{2}+{\left(-x\pm a\right)}^{2}+{\left(-y\pm b\right)}^{2}\right]}^{2}}\text{, \hspace{1em}}-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{,}$$
(11)
$${H}_{\pm 2\pm 2}\left(x,y\right)={J}_{0}\times \frac{{\rho}^{\left|\pm 2\right|}{\rho}^{\left|\pm 2\right|}\rho {L}^{2}}{{\left[{L}^{2}+{\left(x\pm 2a\right)}^{2}+{\left(y\pm 2b\right)}^{2}\right]}^{2}}\text{,}$$
(12)
$$-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{.}$$
(13)
$${H}_{0}\left(x,y\right)={\displaystyle \sum _{j=-\infty}^{\infty}{\displaystyle \sum _{i=-\infty}^{\infty}}}\frac{{J}_{0}{\rho}^{\left|i\right|+\left|j\right|}{L}^{2}}{{\left\{{L}^{2}+{\left[{\left(-1\right)}^{i}x+ia\right]}^{2}+{\left[{\left(-1\right)}^{j}y+jb\right]}^{2}\right\}}^{2}}\text{, \hspace{1em}}-\frac{a}{2}\le x\le \frac{a}{2}\text{,}-\frac{b}{2}\le y\le \frac{b}{2}\text{,}$$
(14)
$${H}_{G1}\left(x,y\right)={H}_{0}\left(x-P,y-Q\right)\text{,}$$
(15)
$${H}_{G2}\left(x,y\right)={H}_{0}\left(x+P,y+Q\right)\text{,}$$
(16)
$${H}_{R}\left(x,y\right)={H}_{0}\left(x-P,y+Q\right)\text{,}$$
(17)
$${H}_{B}\left(x,y\right)={H}_{0}\left(x+P,y-Q\right)\text{,}$$
(18)
$${H}_{T}\left(x,y\right)={H}_{G1}\left(x,y\right)+{H}_{G1}\left(x,y\right)+{H}_{R}\left(x,y\right)+{H}_{B}\left(x,y\right)\text{.}$$
(19)
$$Ur\text{+}=\left(\frac{\text{Maximum}{\left[{H}_{T}\left({x}_{l},{y}_{l}\right)\right]}_{l=\text{10,11,12,13}}}{\text{Average}{\left[{H}_{T}\left({x}_{l},{y}_{l}\right)\right]}_{l=\text{1,2, \hspace{0.17em}}\dots \text{\hspace{0.17em} ,9}}}-1\right)\times 100\%\text{,}$$
(20)
$$Ur\text{\u2212}=\left(\frac{\text{Minimum}{\left[{H}_{T}\left({x}_{l},{y}_{l}\right)\right]}_{l=\text{10,11,12,13}}}{\text{Average}{\left[{H}_{T}\left({x}_{l},{y}_{l}\right)\right]}_{l=\text{1,2, \hspace{0.17em}}\dots \text{\hspace{0.17em} ,9}}}-1\right)\times 100\%\text{,}$$
(21)
$${X}_{m}\left(x,y\right)=k{H}_{m}\left(x,y\right)\times {\displaystyle \sum _{\lambda}{\beta}_{m}}\left(\lambda \right){S}_{m}\left(\lambda \right)\overline{x}\left(\lambda \right)\mathrm{\Delta}\lambda \text{,}$$
(22)
$${Y}_{m}\left(x,y\right)=k{H}_{m}\left(x,y\right)\times {\displaystyle \sum _{\lambda}{\beta}_{m}}\left(\lambda \right){S}_{m}\left(\lambda \right)\overline{y}\left(\lambda \right)\Delta \lambda \text{, \hspace{1em}}m=R,G1,G2,B\text{,}$$
(23)
$${Z}_{m}\left(x,y\right)=k{H}_{m}\left(x,y\right)\times {\displaystyle \sum _{\lambda}{\beta}_{m}}\left(\lambda \right){S}_{m}\left(\lambda \right)\overline{z}\left(\lambda \right)\mathrm{\Delta}\lambda \text{,}$$
(24)
$${X}_{w}\left(x,y\right)={X}_{R}\left(x,y\right)+{X}_{G1}\left(x,y\right)+{X}_{G2}\left(x,y\right)+{X}_{B}\left(x,y\right)\text{,}$$
(25)
$${Y}_{w}\left(x,y\right)={Y}_{R}\left(x,y\right)+{Y}_{G1}\left(x,y\right)+{Y}_{G2}\left(x,y\right)+{Y}_{B}\left(x,y\right)\text{,}$$
(26)
$${Z}_{w}\left(x,y\right)={Z}_{R}\left(x,y\right)+{Z}_{G1}\left(x,y\right)+{Z}_{G2}\left(x,y\right)+{Z}_{B}\left(x,y\right)\text{,}$$
(27)
$$u\prime \left(x,y\right)=\frac{4{X}_{w}}{{X}_{w}+15{Y}_{w}+3{Z}_{w}}\text{,}$$
(28)
$$v\prime \left(x,y\right)=\frac{9{X}_{w}}{{X}_{w}+15{Y}_{w}+3{Z}_{w}}\text{.}$$
(29)
$$\Delta u\prime v\prime ={\left[{({u}_{1}^{\prime}-{u}_{0}^{\prime})}^{2}+{\left({v}_{1}^{\prime}-{v}_{0}^{\prime}\right)}^{2}\right]}^{1/2}\text{,}$$
(30)
$$\Delta {E}_{ab}*\left(x,y\right)=200{\left[{\left(\Delta L*\right)}^{2}+{\left(\Delta a*\right)}^{2}+{\left(\Delta b*\right)}^{2}\right]}^{1/2}\text{,}$$
(31)
$$L*=116{\left(\frac{{Y}_{w}}{{Y}_{n}}\right)}^{1/3}-16\text{,}$$
(32)
$$a*=500\left[{\left(\frac{{X}_{w}}{{X}_{n}}\right)}^{1/3}-{\left(\frac{{Y}_{w}}{{Y}_{n}}\right)}^{1/3}\right]\text{,}$$
(33)
$$b*=200{\left[{\left(\frac{{Y}_{w}}{{Y}_{n}}\right)}^{1/3}-{\left(\frac{{Z}_{w}}{{Z}_{n}}\right)}^{1/3}\right]}^{1/2}\text{,}$$