Solution of radiation and diffraction problems with axial symmetry by the method of parabolic equation

As a rule, problems of radiation and diffraction on bodies of revolution of large electrical size are solved by the method of integral equations, since in this case the original boundary value problem can be reduced to a system of two Fredholm integral equations. On the other hand, for problems with axial symmetry, one can use the factorization method for the Helmholtz equation in cylindrical coordinates. This approach leads to pair of parabolic equations for waves propagating in opposite directions. Thus, the parabolic equation method, which has demonstrated its high efficiency in solving scalar problems of radiation, diffraction and propagation of radio waves, can be generalized to the case of problems with axial symmetry. This paper presents a computationally efficient algorithm for solving a parabolic equation in cylindrical coordinates based on the calculation of the direct and inverse Hankel transform. The accuracy of the presented algorithm was estimated using the example for a perfectly conducting sphere and a truncated Gutman lens. It is shown that for these problems the accuracy of the far-field calculation is comparable with rigorous solutions. The computational capabilities of the method are demonstrated using the example of optical diffraction on a spherical water drop. The considered method can be successfully used to calculate the propagation of axisymmetric wave beams, including vortex beams, which are currently being actively studied. Since all wave beams have a very small diffraction divergence, the parabolic equation method is very well suited for their study. By replacing the Hankel transform with the Fourier transform, we obtain well-known algorithms for calculating the field strength near the Earth. Thus, this paper presents a generalization of the parabolic equation method to the case of problems with axial symmetry. The computational efficiency of the presented algorithms is due to the fact that in modern mathematical software programs all matrix operations are performed very quickly.