Transition between paraxial and Helmholtz Fourier spaces.

Different correction methods for paraxial solutions have been used when such solutions extend out of the paraxial regime. Mapping functions play a fundamental role in them. This paper analyzes how to translate directions of paraxial wavevectors into Helmholtz wavevectors. Two possible mappings that are commonly used are examined, and a new mapping is proposed. The three mappings are classified in the framework of the full wave correction schemes.

Angular spectral framework to test full corrections of paraxial solutions: erratum.

In our previous article [J. Opt. Soc. Am. A32, 1236 (2015)JOAOD60740-323210.1364/JOSAA.32.001236] there is an issue concerning the comparison of plane wave spectrum solutions of paraxial and Helmholtz equations. We compared the angular plane wave spectrum of Helmholtz solutions with the plane wave spectrum of the paraxial solutions in terms of normalized projections of paraxial wave vectors. We show that the proper comparison of plane wave spectra must be done in terms of angles. The results presented in our previous work are corrected accordingly. The most important change is that Wünsche's T2 operator leads to a valid method.

Angular spectral framework to test full corrections of paraxial solutions.

Different correction methods for paraxial solutions have been used when such solutions extend out of the paraxial regime. The authors have used correction methods guided by either their experience or some educated hypothesis pertinent to the particular problem that they were tackling. This article provides a framework so as to classify full wave correction schemes. Thus, for a given solution of the paraxial wave equation, we can select the best correction scheme of those available. Some common correction methods are considered and evaluated under the proposed scope. Another remarkable contribution is obtained by giving the necessary conditions that two solutions of the Helmholtz equation must accomplish to accept a common solution of the parabolic wave equation as a paraxial approximation of both solutions.

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

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.