New treatment of the focusing method and tomography of the refractive index distribution of inhomogeneous optical components

Abstract

A new mathematical formulation of the focusing method connecting the refractive index distribution of inhomogeneous optical components to the transversally transmitted light distribution is presented. This formulation has the following advantages: the characteristic singularity occurring in the focusing method is avoided; a single numerical integration is used instead of a double one, ensuring a faster data processing; the refractionless approximation is introduced only in the final step of calculations, leading to smaller errors; and the avoidance of a residual logarithmic singularity. An inverse functional transform connecting the intensity distribution in the near field to the refractive index distribution inside the object is deduced and used to obtain analytical test functions. The theoretical results are supported by computer simulations and experiments with gradient-index rods and optical preforms, which have shown improvements in the accuracy, stability, and speed of the method. Moreover, this mathematical formulation for the axisymmetric objects is generalized to a tomographic formulation for nonsymmetric objects using the inverse Radon transform. An efficient algorithm and suitable computer simulations are presented for this case.