Depth profiles in confocal optical microscopy: a simulation approach based on the second Rayleigh-Sommerfeld diffraction integral.

A method is presented for recovering the intensity depth profile, by confocal optical microscopy, in transparent and amorphous samples with low scattering. The response function of a confocal Raman microscope has been determined by using the second Rayleigh-Sommerfeld diffraction integral and scalar wave optics within paraxial approximation, taking into account the refractive index mismatch between the sample and the medium surrounding the objective lens. An iterative multi-fitting-scheme, based on the conjugate gradient method and Brent algorithm, allowed to fit several depth profile curves simultaneously and retrieve the beam waist, the signal amplitude and the position of the sample surface. The reliability and accuracy of the theoretical procedure has been investigated through comparison with experimental measurements of the Raman depth profiles for different pinhole diameters. The model is shown to provide accurate description of the effect of the mismatch of the refractive index and of the dependence of the Raman signal on the depth with discrepancies lower than 3%. This procedure constitutes a first step towards the development of a manageable theoretical framework, amenable to a relatively simple numerical implementation, for the solution of the 'inverse' problem of finding the correct reconstruction of unknown profiles of chemical species within the sample, starting from experimental information gathered from micro-Raman depth profiling.