Differences between the narrow-angle and wide-angle propagators in the split-step Fourier solution of the parabolic wave equation

For tropospheric electromagnetic propagation, Maxwell´s equations can be reduced to a parabolic wave equation, which is solved by marching over range steps. In each step, the solution is split into a product of three operators. The first and third account for atmospheric and surface variation, while the center operator propagates the field as though in vacuum. This center operator is the object of interest here. Older versions of the method used the narrow-angle propagator, while more recent versions use the wide-angle propagator. It was thought that the wide-angle propagator was entirely superior to the narrow-angle propagator, but some artifacts observed in experiments have led to the present investigation. The two propagators are examined numerically and analytically and found to exhibit subtle differences at large angles from the horizontal. This has required modifications to the way in which sources are created for starting the split-step solution. The narrow- and wide-angle propagators are also compared on two problems with analytic solutions to quantify the improvement of the wide-angle over the narrow-angle