Non-destructive analysis of concentration profiles in turbid media using micro-spatially offset Raman spectroscopy: A physical model

We introduce an analytical physical description of micro-spatially offset Raman spectroscopy (micro-SORS) experiments in turbid samples with a special emphasis on its use for studying the penetration depth of chemical species (agents) into a matrix. Such samples exhibit varying concentration gradients with depth and reflect typical situations encountered in Cultural Heritage framework. The experiments aim at yielding the variations (trends) of the Raman intensity ratio of the agent and matrix marker bands, (Formula presented.), with increasing micro-SORS defocusing distance (Formula presented.). The physical model shows that (Formula presented.) can be expressed as a function of the concentration profiles of the analytes. The analysis makes use of Monte Carlo simulation results of the propagation of photons in turbid media. Despite the introduced approximations, the obtained formulas provide an insightful tool that allows justifying the observed trends and facilitates the interpretation of micro-SORS data and systematic comparison among different samples. Basic comparison with previously published experimental micro-SORS data indicates good consistency. The obtained relationships could also be useful for a semi-quantitative estimate of concentration profiles in unknown samples.