## Abstract

Recent studies have shown that the slope of logarithmic scattering spectroscopy of a turbid medium is related to the sizes of the scattering particles within the turbid medium. Mie theory can be used to generate a logarithmic plot of the reduced-scattering coefficient versus wavelength. According to Nilsson *et al*. [Appl. Opt. **37**, 1256 (1998)], the slope value of a linear fit of the logarithmic scattering spectroscopy between 600 and 1050 nm can be used for direct determination of particle size. We performed similar calculations using the Rayleigh-Gans approximation and obtained an analogous overall shape with additional sinusoidal features. Our calculations indicate a possible relationship between the slope and the particle size when the size is used to calculate the slope, namely, in the forward calculation. However, because of the sinusoidal pattern, the inverse calculation to obtain the particle size from the slope may be applied only for particles with a radius of <0.13 μm in combination with 650–1050-nm light. Caution should be exercised when inverse calculation is performed to determine the scattering particle sizes in the range of radii >0.13 μm, with the slope of logarithmic scattering spectroscopy within 650–1050 nm.

© 2003 Optical Society of America

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### Equations (8)

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(1)
$${\mathrm{\mu}}_{s}^{\prime}=\mathrm{\rho}\left[{Q}_{\mathrm{sca}}\left(1-g\right)\mathrm{\pi}{r}^{2}\right]=\mathrm{\rho}{Q}_{\text{sca}}^{\prime}\mathrm{\pi}{r}^{2}.$$
(2)
$$\frac{{\mathrm{\mu}}_{s}^{\prime}\left(\mathrm{\lambda}\right)}{{\mathrm{\mu}}_{s}^{\prime}\left({\mathrm{\lambda}}_{0}\right)}=\frac{\mathrm{\rho}{Q}_{\text{sca}}^{\prime}\left(\mathrm{\lambda}\right)\mathrm{\pi}{r}^{2}}{\mathrm{\rho}{Q}_{\text{sca}}^{\prime}\left({\mathrm{\lambda}}_{0}\right)\mathrm{\pi}{r}^{2}}=\frac{{Q}_{\text{sca}}^{\prime}\left(\mathrm{\lambda}\right)}{{Q}_{\text{sca}}^{\prime}\left({\mathrm{\lambda}}_{0}\right)}.$$
(3)
$$ln\left({\mathrm{\mu}}_{s}^{\prime}\right)=ln\left(k\right)+nln\left(\mathrm{\lambda}\right).$$
(4)
$$n=-1109.5{r}^{3}+341.67{r}^{2}-9.369r-3.9359,\mathrm{for}r0.23\mathrm{\mu}\mathrm{m},$$
(5)
$$n=23.909{r}^{3}-37.218{r}^{2}+19.534r-3.965,\mathrm{for}0.23\le r\le 0.6\mathrm{\mu}\mathrm{m}.$$
(6)
$$\mathrm{\sigma}{\prime}_{s}\left(\mathrm{\lambda}\right)=\frac{9\mathrm{\pi}{r}^{2}}{64{x}^{2}}{\left|\frac{{m}^{2}-1}{{m}^{2}+2}\right|}^{2}{\int}_{0}^{\mathrm{\pi}}{\left(sinu-ucosu\right)}^{2}\times \frac{\left(1+{cos}^{2}\mathrm{\theta}\right)sin\mathrm{\theta}\left(1-cos\mathrm{\theta}\right)}{{sin}^{6}\left(\mathrm{\theta}/2\right)}\mathrm{d}\mathrm{\theta}=\frac{9}{256\mathrm{\pi}}{\left(\frac{\mathrm{\lambda}}{{n}_{\mathrm{ex}}}\right)}^{2}{\left|\frac{{m}^{2}-1}{{m}^{2}+2}\right|}^{2}{\int}_{0}^{\mathrm{\pi}}{\left(sinu-ucosu\right)}^{2}\times \frac{\left(1+{cos}^{2}\mathrm{\theta}\right)sin\mathrm{\theta}\left(1-cos\mathrm{\theta}\right)}{{sin}^{6}\left(\mathrm{\theta}/2\right)}\mathrm{d}\mathrm{\theta},$$
(7)
$$u=2xsin\left(\frac{\mathrm{\theta}}{2}\right)=2\left(\frac{2\mathrm{\pi}{\mathit{rn}}_{\mathrm{ex}}}{\mathrm{\lambda}}\right)sin\left(\frac{\mathrm{\theta}}{2}\right),$$
(8)
$${Q}_{\text{sca}}^{\prime}\left(\mathrm{\lambda}\right)=\frac{\mathrm{\sigma}{\prime}_{s}\left(\mathrm{\lambda}\right)}{\mathrm{\pi}{r}^{2}}.$$