Transition between free-space Helmholtz equation solutions with plane sources and parabolic wave equation solutions

Abstract

The slowly varying envelope approximation is applied to the radiation problems of the Helmholtz equation with a planar single-layer and dipolar sources. The analyses of such problems provide procedures to recover solutions of the Helmholtz equation based on the evaluation of solutions of the parabolic wave equation at a given plane. Furthermore, the conditions that must be fulfilled to apply each procedure are also discussed. The relations to previous work are given as well.

Angular spectral framework to test full corrections of paraxial solutions

“…The results have been applied to three correction methods in their general form and we have shown that, under these circumstances, the method that consists of taking the paraxial wave at a transverse plane to the axial direction as the Dirichlet boundary condition of the Helmholtz equation is worse, in general, than the other full correction schemes considered. This result is not new since there were previous criticisms to this method [38,39].…”

“…Wünsche's T 2 operator [3] defines the correction scheme by using the paraxial wave at a transversal plane to the axial direction as a Neumann boundary condition of the Helmholtz equation. This method has barely been applied; although in some articles [38,39] it is recommended instead of the scheme given by the T 1 operator. There is another full correction scheme proposed by Davis [7], but to our knowledge it has not yet been applied as a full wave correction method.…”