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We derive and validate an analytical model that describes the migration of Raman scattered photons in two-layer diffusive media, based on the diffusion equation in the time domain. The model is derived under a heuristic approximation that background optical properties are identical on the excitation and Raman emission wavelengths. Methods for the reconstruction of two-layer Raman spectra have been developed, tested in computer simulations and validated on tissue-mimicking phantom measurements data. Effects of different parameters were studied in simulations, showing that the thickness of the top layer and number of detected photon counts have the most significant impact on the reconstruction. The concept of quantitative, mathematically rigorous reconstruction using the proposed model was finally proven on experimental measurements, by successfully separating the spectra of silicone and calcium carbonate (calcite) layers, showing the potential for further development and eventual application in clinical diagnostics.

We present a new technique, frequency offset Raman spectroscopy (FORS), to probe Raman spectra of diffusive media in depth. The proposed methodology obtains depth sensitivity exploiting changes in optical properties (absorption and scattering) with excitation wavelengths. The approach was demonstrated experimentally on a two-layer tissue phantom and compared with the already consolidated spatially offset Raman spectroscopy (SORS) technique. FORS attains a similar enhancement of signal from deep layers as SORS, namely 2.81 against 2.62, while the combined hybrid FORS-SORS approach leads to a markedly higher 6.0 enhancement. Differences and analogies between FORS and SORS are discussed, suggesting FORS as an additional or complementary approach for probing heterogeneous media such as biological tissues in depth.

A set of time-domain analytical forward solvers for Raman signals detected from homogeneous diffusive media is presented. The time-domain solvers have been developed for two geometries: the parallelepiped and the finite cylinder. The potential presence of a background fluorescence emission, contaminating the Raman signal, has also been taken into account. All the solvers have been obtained as solutions of the time dependent diffusion equation. The validation of the solvers has been performed by means of comparisons with the results of "gold standard" Monte Carlo simulations. These forward solvers provide an accurate tool to explore the information content encoded in the time-resolved Raman measurements.