## Abstract

We investigated, both analytically and numerically, the irradiance formation of an asymmetrically located Lambertian light source in hollow straight light pipes with square and circular shapes. The uniform irradiance distribution in a square light pipe and hot-spot localization in a circular light pipe were examined and determined semianalytically. Typical factors of influence, such as light-pipe length, width, and source size, were identified with extensive simulation. When the ratio of light-pipe length and width was less than 0.5, the deviation from uniformity could be more than 20%. But once the source size was large enough (approximately half of the incident port), such that the Lambertian characteristics of the source dominated the irradiance distribution, the uniformity deviation was reduced. Furthermore, a quantity of root-mean-square circular differences was defined in order to identify the shape deformation of the light pipe; it was found that the peak value of the hot spot decreased exponentially with the deformation scale. The influence of nonperfect reflectivity of the pipe wall on irradiance formation was also examined for a square light pipe; when the reflectivity is larger than 90%, the difference in uniformity is less than 10% and uniform irradiance remains, provided that the ratio of light-pipe length and width is larger than 1; even the source is located asymmetrically.

© 2006 Optical Society of America

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### Equations (10)

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(1)
$$h\left(\theta \right)=k+{(-1)}^{m}(L\phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta -mD),$$
(2)
$$E\left({h}_{0}\right)=\sum _{i=1}^{\infty}\phantom{\rule{0.2em}{0ex}}\mathrm{cos}\phantom{\rule{0.2em}{0ex}}{\theta}_{i}.$$
(3)
$$h(\theta ,d)=k+{(-1)}^{{m}^{\prime}}(L\phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta +d-{m}^{\prime}D),$$
(4)
$$h(-\theta ,d)=k+{(-1)}^{{m}^{\u2033}}(\mid L\phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\theta -d\mid -{m}^{\u2033}D),$$
(5)
$$h(-\theta ,d)=D-h(\theta ,D-d),$$
(6)
$$E{\left(h\right)}_{\text{full}\phantom{\rule{0.3em}{0ex}}\text{angle}}=E{\left(h\right)}_{+\theta}+E{\left(h\right)}_{-\theta},$$
(7)
$$\lfloor {P}_{x}(\theta ,\varphi ),{P}_{y}(\theta ,\varphi )\rfloor =[h\left(\theta \right)\mathrm{sin}\phantom{\rule{0.2em}{0ex}}\varphi ,h\left(\theta \right)\mathrm{cos}\phantom{\rule{0.2em}{0ex}}\varphi ].$$
(8)
$$E\left(R\right)=\frac{\text{total}\phantom{\rule{0.3em}{0ex}}\text{flux}\phantom{\rule{0.3em}{0ex}}\text{of}\phantom{\rule{0.3em}{0ex}}\text{one}\phantom{\rule{0.3em}{0ex}}\text{circle}}{\text{total}\phantom{\rule{0.3em}{0ex}}\text{area}\phantom{\rule{0.3em}{0ex}}\text{of}\phantom{\rule{0.3em}{0ex}}\text{one}\phantom{\rule{0.3em}{0ex}}\text{circle}}=\frac{\left(\frac{F}{{N}_{d}}\right){N}_{s}}{\pi {R}^{2}-\pi {(R-\Delta D)}^{2}}=\frac{\frac{F}{D\u2215\Delta D}\frac{2\pi}{\Delta \varphi}}{2\pi R\Delta D-\pi \Delta {D}^{2}}=\left(\frac{F}{D\Delta \varphi}\frac{1}{R}\right){(1-\frac{\Delta D}{2R})}^{-1}\approx \left(\frac{F}{D\Delta \varphi}\frac{1}{R}\right)[1+\frac{\Delta D}{2R}-{\left(\frac{\Delta D}{2R}\right)}^{2}+\cdots ].$$
(9)
$$\delta =\raisebox{1ex}{$\left[\frac{1}{n}\sum _{i=1}^{n}\mid {E}_{i}-\overline{E}\mid \right]$}\!\left/ \!\raisebox{-1ex}{$\overline{E}$}\right.,$$
(10)
$$\Delta =\frac{1}{2\pi}\sqrt{{\int}_{0}^{2\pi}{({D}_{m}\left(\theta \right)-{R}_{s})}^{2}\mathrm{d}\theta},$$