Photon density waves scattered from cylindrical inhomogeneities: theory and experiments

We present an analytical solution for the scattering of diffuse photon density waves from an infinite circular, cylindrical inhomogeneity embedded in a homogeneous highly scattering turbid medium. The analytical solution, based on the diffusion approximation of the Boltzmann transport equation, represents the contribution of the cylindrical inhomogeneity as a series of modified Bessel functions integrated from zero to infinity and weighted by different angular dependencies. This series is truncated at the desired precision, similar to the Mie theory. We introduce new boundary conditions that account for specular reflections at the interface between the background medium and the cylindrical inhomogeneity. These new boundary conditions allow the separate recovery of the index of refraction of an object from its absorption and reduced scattering coefficients. The analytical solution is compared with data obtained experimentally to evaluate the predictive capability of the model. Optical properties of known cylindrical objects are recovered accurately. However, as the radius of the cylinder decreases, the required measurement signal-to-noise ratiorapidly increases. Because of the new boundary conditions, an upperlimit can be placed on the recovered size of cylindrical objects with radii below 0.3 cm if they have a substantially different index of refraction from that of the background medium.