Penetration of Light into Multiple Scattering Media: Model Calculations and Reflectance Experiments. Part I: The Axial Transfer

Abstract

The article presents two general equations of radiation penetration into layers of diffuse reflectors. One of the equations describes the depth origins of reflection, the other the depth profiles of absorption. The equations are evaluated within the theory of radiative transfer applying various degrees of analytical approximations and Monte Carlo simulations. The data are presented for different scattering and absorption coefficients, arbitrary layer thicknesses, collimated and diffused irradiation, and anisotropic forward scattering. The calculated mean depths of reflection are always lower than the mean depths of absorption. For nearly non-absorbing layers, the mean depths of absorption are about one third of the physical layer thickness. In contrast, penetration saturates for strong absorbers at very low depth levels. From the simulated data, methods are derived for the determination of the penetration depth from reflectance and transmittance data of thin layers or from radially diffused reflectance profiles upon spot irradiation. The methods are experimentally verified for a series of metal oxide powders with particle sizes ranging from much smaller to much larger than the wavelength of irradiation and for microcrystalline cellulose stained with different concentrations of an organic dye.

Pigmentation of White, Brown, and Green Chicken Eggshells Analyzed by Reflectance, Transmittance, and Fluorescence Spectroscopy

“…However, the non‐uniform axial and lateral distribution of the proteins, as well as the lack of reliable transmittance data (the T‐values are too low for quantitative evaluation, see Figure and Figure ) make a full analysis almost impossible. In a simple approach, we treated the shell as semi‐infinite layer with an optical sampling depth of 5–10 μm, and extrapolated the scattering coefficient from Table . The results are presented in Table .…”

“…The optical parameters of the shells were calculated from the spectroscopic data with the continuum model of radiative transfer, which is an extension of Beer's law to three‐dimensionally multiple scattering systems. In order to reduce the mathematical effort for the description of the photo‐stationary state, we used the random‐walk (RW) approach, and split the incident radiation flux into a large number (e. g. one million) of bunched beams (“photons”), and recorded one after the other their stochastic fates through the sample until reflection, transmission or absorption. The sum of all experiments yields the photo‐stationary state, with the disadvantage of lacking analytical presentation, but the great advantage of applicability to systems with axial and radial material gradients as, e. g., in an eggshell.…”

“…According to the graphical TOC, we apply diffuse reflectance, diffuse transmittance and fluorescence spectroscopy for the in‐shell analysis of the optical concentrations and axial concentration profiles of PPIX, BV, and proteins in the multiple scattering shell. The sampling depth of reflectance reaches up to one third of the total layer thickness but decreases with increasing absorption . Transmittance probes the whole layer depth with slight preference of the central region.…”

“…The sampling depth of reflectance reaches up to one third of the total layer thickness but decreases with increasing absorption. [44] Transmittance probes the whole layer depth with slight preference of the central region. For a full analysis, we acquire data with irradiation from both surface sides and, when indicated, after mechanical ablation of the outer shell regions.…”

Variation in the Transmission Near-Infrared Signal with Depth in Turbid Media

“…These findings are also consistent with a recent study of the penetration of light into multiple scattering media. 27 Monte Carlo simulations were used to calculate the probability density (equal to the residence time of a photon at a given depth, relative to the residence time of a normally incident photon in a scatter-free layer) as a function of depth, with the maximum probability density observed at the center of the layer.…”