A Finite Hankel Algorithm For Intense Optical Beam Propagation In Saturable Medium

Abstract

Many physical problems, especially light-propagation 1-6 , that involve the Laplacian operator, are naturally connected with Fourier or Hankel transforms (in case of axial symmetry), which both remove the Laplacian term in the transformed space. Sometimes the analytical calculation can be handled at its end, giving a series or an integral representation of the solution. Otherwise, an analytical pre-treatment of the original equation may be done, leading to numerical computational techniques such as in Refs. 3,4,5 as opposed to self-adaptive stretching and rezoning techniques6, which do not use Fourier or Hankel transforms. We will present here some basic mathematical properties of infinite and finite Hankel transform, their connection with physics and their adaptation to numerical calculation. The finite Hankel transform is well suited to numerical computation, because it deals with a finite interval, and the precision of the calculation can be easily controlled by the number of zeros of Jo(x) to be taken. Moreover, we use a special quadrature formula which is well connected to integral conservation laws. The inconvenience of having to sum a series is reduced by the use of vectorized computers, and in the future will be still more reduced with parallel processors. A finite-Hankel code has been performed on CRAY-XMP in order to solve the propagation of a CW optical beam in a saturable absorber. For large diffractions or when a very small radial grid is required for the description of the optical field, this FHT algorithm has been found to perform better than a direct finite-difference code.