A unified and generalized formulation for the complexspatial point source beam solutions of the Helmholtz wave equation in the spatial and the Fourier-spectral domains for both, 3D and 2D scenarios, is presented in this paper. This general formulation is based on the description of the solutions in terms of complex distances and complex angles, emphasizing the physical meaning of these complex quantities and the relationship between spatial and spectral representations.

The well known 2D complex-point-source beam presents undesired field characteristics. This paper proposes some combinations of simple complex beams which allow the construction of well behaved fields.

The definition of a complex polar coordinate system that unifies complex distances and complex angles together with their relations with the real observation space is presented in this paper. Its utility is shown through some applications to electromagnetic problems: (i) new solutions are obtained from well-known solutions in real polar coordinates, (ii) the parameterization of the real space in terms of the complex polar coordinates helps to understand the physical behaviour of those solutions, and (iii) the results provide a better physical insight of the complex distances and angles.

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.

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