Home | Survey | Topics | Index | References | Dictionary | Contributing | Gallery | Community
| Transmission function in the small-angle approximation .. : SAA and exact solutions | Parent topic |
Fig.2. Top panel: Dependence of the normalized transmission function, T, on the observation angle, γ, for various optical thicknesses, τ = 0.1 (black), 1 (red), 2 (blue), 5 (green), and 10 (brown) (bottom to top) according to the small-angle approximation solution (Eq. 6 in Transmission function in the small-angle approximation for a medium illuminated at normal incidence), Ta (deep-colored lines), and the exact numerical solution, Te (faint-colored lines), of the RTE for a turbid medium with spherical and homogeneous particles having a gamma PSD (parameters: α = 6, Deff = 8 µm) and a refractive index of 1.52 - 0.008i relative to that of the surrounding medium. Calculations have been performed for a wavelength of λ = 0.5 µm. The phase function and other relevant optical properties of the particles were calculated according to Mie theory. The exact numerical solution is described by Rozanov and Kokhanovsky (2006). Note a significant flattening of the transmission function with the increasing optical thickness, τ. The values at 0° of all exact normalized transmission functions are equal to unity, those of the approximate normalized functions are respectively 1.001927, 1.000318, 1.000408, 1.000728, and 1.000317 for τ = 0.1, 1, 2, 5, and 10. The absolute values of the exact transmission functions at 0° are respectively 33.9574, 151.239, 124.015, 22.8626, and 0.79291. Bottom panel: Relative error of the approximate solution for the transmission function at the optical thicknesses of τ = 0.1 (black), 1 (red), 2 (blue), 5 (green), and 10 (brown). Small oscillations of the errors are most likely due to a limited accuracy of the exact numerical solution of the RTE, which is rarely better than 0.1 to 1%. See also a tab-delimited data file.
| CITATION: Kokhanovsky A. A. 2007. Transmission function in the small-angle approximation (SAA) for a medium illuminated at normal incidence: SAA and exact solutions (www.tpdsci.com/Tpc/RTSAANmlTsmCmp.php). In: Top. Part. Disp. Sci. (www.tpdsci.com). |
HISTORY: Published: 14-Feb-2007 Modified: 05-Feb-2007 Peer-reviewed: PENDING |
| Copyright 2005-2008 MJC Optical Technology. All rights reserved. | Terms of use | Menu |