Parameters Governing Deviations from the Beer-Lambert-Bouger Law for Direct Transmission

The study of transmission of radiation through discrete random media has applications in both atmospheric science and nuclear physics [1,2]. Assuming a volume populated with uncorrelated, discrete, perfectly absorbing obstacles which is otherwise optically transparent, the Beer-Lambert-Bouguer (BLB) law of exponential attenuation provides a ready model of radiation intensity I as a function of penetration depth z, namely I(z) = I0 exp(-cσz). Here, c represents the concentration of absorbers per unit volume, while σ is the effective cross section of each absorber.

In simulation, ballistic photon models of the radiative transfer process adhere to the BLB law under the assumptions stated above; however, as shown in [3], even small deviations from the assumption of uncorrelated particle placement (that is, deviations from perfect randomness) can result in large deviations from this expected attenuation. The computational study completed in [3] further suggests that the deviations from the BLB law persist even if the spatial correlations are at scales much smaller than the characteristic lengths associated with the transmission. Thus it would appear that two scales are of physical relevance: a measure of correlation length / cluster size within the volume, and the quantity (cσ)-1, the mean free path length.

An easily parameterized relationship connecting deviations from BLB to statistical properties of the absorbers in a random medium would be of great utility in various fields. Recent efforts include attempts to describe deviations as a function of pair-correlation (or fractal dimension, or Stokes number), the size of the volume, and the classical optical depth. However, our study [4], also using a ballistic photon model of the transfer process, indicates that there are additional physical scales which must be taken in to consideration when trying to define such a relationship. In this presentation, we will describes the results from [4] and their implications, as well as discuss on-going numerical simulations.

1. Kostinski, AB. On the extinction of radiation by a homogeneous but spatially correlated random medium: reply to comment. J. Opt Soc Am A 2002;19:2521-5.

2. Davis AB, Mineev-Weinstein MB. Radiation propagation in random media: from positive to negative correlations in high-frequency fluctuations. J Quant Spec Radiat Transfer 2011;112:632-45.

3. Shaw, RA, Kostinski AB, Lanterman DD. Super-exponential extinction of radiation in a negatively correlated random medium. J Quant Spec Radiat Transfer 2002;75:13 – 20.

4. Larsen ML, Clark AS. On the link between particle size and deviations from the Beer-Lambert-Bouguer law for direct transmission. J Quant Spec Radiat Transfer 2014;133:646 – 651.