Analytical Solution to the Depth-of-Origin Profile of Transmission Raman Spectroscopy in Turbid Media Based on the Kubelka-Munk Model.

An analytical formula to the depth-of-origin profile of transmission Raman spectroscopy in turbid media was derived from the one-dimensional (1D) Kubelka-Munk model. The depth-of-origin profile of the transmitted Raman is proportional to the excitation intensity profile and the transmittance profile, which are two functions of similar forms. The effect of scattering, absorption, and signal-enhancing reflectors are incorporated into the formula. Depth-of-origin profile of a model sample was measured at better than 0.2 mm resolution and fits the formula reasonably well. Conical reflective cavities placed at the front and/or back of the sample enhanced the signal significantly; the relationship among the enhancement functions is verified by the formula. Optical parameters derived from the fitting are compared to theoretical value predicted by optical ray tracing and direct measurements; discrepancies are related to deficiency of the 1D Kubelka-Munk model.