Optical remote sensing inside an inhomogeneous axisymmetric medium: the absorption field measurement

It is shown that the absorption field inside an inhomogeneous, rotationally symmetric medium with a spatially variable refractive index can be reconstructed by means of a tomographic technique. The classic Abel transform is extended to non-Euclidean optical media. The optical behavior of such a medium is described and, provided that the product of the refractive index with the radial distance is a monotonic function, an exact inverse formula is found. Both a numerical and an analytical test on a phantom function is carried out to prove the exactness of this formula. In contrast, when the assumption of a monotonic function is not true, it is shown that the reconstruction problem becomes subdeterminate because of the presence of annular regions, known as blind areas, inside of which no curved path reaches an extremum. The spatial localization and the size of these regions are related to the extrema of the index of refraction times the radial distance.