2D Complex Point Source Radiation Problem II.~Complex Beams

Authors: M. J. GONZALEZ MORALES, E. GAGO-RIBAS

Abstract: The complex point source analytic continuation, and the analysis into the spaces of complex distances, angles, and other related magnitudes, may be applied to describe a wide variety of 2D wave propagation problems. This step will be essential in order to establish a complete complex methodology which may be applied to obtain general descriptions of more practical problems involving the scattering of waves under some kind of field incidence. A particular application of the methodology presented in Part I will be used in Part II. The specific problem under analysis in these papers will be the general complex beam solution obtained from the analytical continuation of the real space Green's function, solution of the 2D Helmholtz equation in free space, into the space of complex coordinates. As obtained in Part I, the general complex beam solution turns out a great variety of solutions, each one associated to a specific practical approximation. The complete set of solutions for this particular problem, already presented in Part I, will be analyzed here in detail. The initial representation of the problem in terms of different regions associated to each solution (obtained by parameterizing each approximation into the complex distances and complex angles spaces), will be used here to describe the behavior (amplitude profile, phase fronts, energy phase paths, etc.) for each solution, such as non homogeneous cylindrical waves or pseudo-Gaussian beams, elliptical-phase non homogeneous cylindrical waves, Gaussian beams, etc. The complex representation of important magnitudes such as phase and energy paths will be analyzed here by using some of the mappings presented in Part I. This kind of interpretations will be essential when trying to understand complex rays, complex image sources or complex caustics in later scattering problems. The methodology, described first in Part I and completed now in Part II, should conform a general basis to study other wave propagation problems described in terms of complex source coordinates.

Keywords: complex source, radiation, Gaussian beam, paraxial condition, propagation

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