## Abstract

We have characterized the path length for the differential path-length spectroscopy (DPS)
fiber optic geometry for a wide range of optical properties and for fiber diameters ranging from
$200\text{\hspace{0.17em} \mu m}$ to
$\text{1000 \hspace{0.17em} \mu m}$. Phantom measurements show that the path length is nearly constant for scattering coefficients in the range
$5\text{\hspace{0.17em}}{\text{mm}}^{-1}<{\text{\mu}}_{s}<50\text{\hspace{0.17em}}{\text{mm}}^{-1}$ for all fiber diameters and that the path length is proportional to the fiber diameter. The path length decreases with increasing absorption for all fiber diameters, and this effect is more pronounced for larger fiber diameters. An empirical model is formulated that relates the DPS path length to total absorption for all fiber diameters simultaneously.

© 2008 Optical Society of America

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### Equations (6)

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(2)
$$R={R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left(-\tau {{{\displaystyle \mathbf{\mu}}}_{a}}^{\text{specific}}c\right)\text{,}$$
(3)
$$R={c}_{\text{cal}}\left[\frac{\left(I-{I}_{\text{water}}\right)}{\left({I}_{\text{white}}-{I}_{\mathrm{b}\mathrm{l}\mathrm{a}\mathrm{c}\mathrm{k}}\right)}-\frac{J}{\left({J}_{\text{white}}-{J}_{\text{black}}\right)}\right]\text{,}$$
(4)
$$\tau =-\frac{\text{ln}\left(R/{R}_{0}\right)}{{{{\displaystyle \text{\mu}}}_{a}}^{\text{specific}}c}\text{.}$$
(5)
$$\frac{{\tau}_{\text{model}}}{{d}_{\text{fiber}}}=\frac{{{{\displaystyle c}}_{1}}^{\text{empirical}}}{1+\text{ln}\left(\text{1 \hspace{0.17em}+ \hspace{0.17em}}{{{\displaystyle c}}_{\text{2}}}^{\text{empirical}}{\text{\mu}}_{a}{d}_{\text{fiber}}\right)}\text{.}$$
(6)
$$R={R}_{0}\text{\hspace{0.17em}}\mathrm{exp}\left(-0.95{d}_{\text{fiber}}{{{\displaystyle \text{\mu}}}_{a}}^{\text{specific}}c\right)\text{.}$$