## Abstract

The transmittance of skew rays through metal light pipes is
examined with ray tracing. The transmittance with respect to pipe
length is compared with analytical approximations and with experimental
data. The effects of pipe material, pipe shape, wavelength of the
incident light, distribution of the incident light, and maximum angle
of incidence on transmittance are examined. The transmittance is
shown, in general, not to be exponential with respect to pipe
length. Additionally, the effect on transmittance of elbows and
gaps in a pipe is investigated.

© 1999 Optical Society of America

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### Equations (13)

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(1)
$$\mathbf{E}={\mathbf{E}}_{0}exp\left(i\mathbf{k}\xb7\mathbf{p}\right),$$
(2)
$${r}_{s}=\frac{{cos}^{2}{\mathrm{\theta}}_{i}-\sqrt{\mathrm{\u220a}-{sin}^{2}{\mathrm{\theta}}_{i}}}{{cos}^{2}{\mathrm{\theta}}_{i}+\sqrt{\mathrm{\u220a}-{sin}^{2}{\mathrm{\theta}}_{i}}},{r}_{p}=\frac{\mathrm{\u220a}{cos}^{2}{\mathrm{\theta}}_{i}-\sqrt{\mathrm{\u220a}-{sin}^{2}{\mathrm{\theta}}_{i}}}{\mathrm{\u220a}{cos}^{2}{\mathrm{\theta}}_{i}+\sqrt{\mathrm{\u220a}-{sin}^{2}{\mathrm{\theta}}_{i}}},$$
(3)
$${\mathrm{\theta}}_{r}={cos}^{-1}\left[\frac{\left({\mathbf{k}}_{1}\times {\mathbf{n}}_{1}\right)\xb7\left({\mathbf{k}}_{1}\times {\mathbf{n}}_{2}\right)}{|{\mathbf{k}}_{1}\times {\mathbf{n}}_{1}|{\mathbf{k}}_{1}\times {\mathbf{n}}_{2}|}\right].$$
(4)
$$T=\frac{\sum w|{\mathbf{E}}_{f}{|}^{2}}{\sum w|{\mathbf{E}}_{i}{|}^{2}},$$
(5)
$$T=\xbd\left[1+exp\left(-2q\right)\right]-\frac{{\mathit{qF}}^{2}}{8},q=0.09\sqrt{\frac{\mathrm{\rho}}{\mathrm{\lambda}}}\frac{L}{R}.$$
(6)
$$T=exp\left(-\sqrt{\frac{\mathit{\nu}}{\mathrm{\sigma}}}\frac{L}{2R}\right),$$
(7)
$$T={\left(r_{s}{}^{m}{E}_{s}\right)}^{2}+{\left(r_{p}{}^{m}{E}_{p}\right)}^{2}.$$
(8)
$${E}_{s}\prime ={r}_{s}{E}_{s}cos{\mathrm{\theta}}_{r}+{r}_{p}{E}_{p}sin{\mathrm{\theta}}_{r},$$
(9)
$${E}_{p}\prime ={r}_{p}{E}_{p}cos{\mathrm{\theta}}_{r}-{r}_{s}{E}_{s}sin{\mathrm{\theta}}_{r}.$$
(10)
$$\left[\begin{array}{c}{E}_{s}\\ {E}_{p}\end{array}\right]=Aexp\left(i\mathrm{\delta}\right)\left[\begin{array}{c}cos{\mathrm{\theta}}_{a}cos\mathrm{\u220a}-isin{\mathrm{\theta}}_{a}sin\mathrm{\u220a}\\ sin{\mathrm{\theta}}_{a}cos\mathrm{\u220a}+icos{\mathrm{\theta}}_{a}sin\mathrm{\u220a}\end{array}\right].$$
(11)
$$\mathrm{\u220a}=1-\frac{\mathrm{\omega}_{p}{}^{2}}{{\mathrm{\omega}}^{2}+{\mathrm{\gamma}}^{2}}+i\frac{\mathrm{\omega}_{p}{}^{2}\mathrm{\gamma}}{\mathrm{\omega}\left({\mathrm{\omega}}^{2}+{\mathrm{\gamma}}^{2}\right)},$$
(12)
$$4\mathrm{\pi}{\mathrm{\sigma}}_{0}=\mathrm{\omega}_{p}{}^{2}\mathrm{\tau}.$$
(13)
$$w\left(r\right)=exp\left(\frac{-{r}^{2}}{2{\mathrm{\sigma}}^{2}}\right),$$