‘Complex source’ wavefields: sources in real space

Abstract

We consider a complexified Green function for the 3D Helmholtz operator. This function, G* = exp (ikR*)/R*, where , describes a Gaussian beam propagating along the z-axis. It satisfies a certain inhomogeneous Helmholtz equation in . The corresponding source distribution is a generalized function supported by a 2D surface, , which depends on the branch cut of the square root defining R*. We discuss various choices of the branch cut, and show that the corresponding surface can have a complicated structure. In order to describe the source distribution, we first choose the cut and the corresponding branch of the square root, and then regularize a certain integral with a power-type singularity. The paraxial and far-field asymptotic representations are presented for G*.

Complex Point Source for the 3d Laplace Operator

“…The total regularized potential is the sum of the results (18) and (20). Finally, the real potential is obtained by making the limit as p → 1:…”

“…Mahillo-Isla et al [19], give a physical solution to the equivalent sources on the disk itself based on the evaluation of the field. Finally, Tagirdzhanov et al consider in [20] the complexified Green's function of the Helmholtz equation and the field sources that create that solution. In [21], they deal with the singularity that appears in the rim of the disk by covering it by a circular torus.…”

“…Finally, Tagirdzhanov et al consider in [20] the complexified Green's function of the Helmholtz equation and the field sources that create that solution. In [21], they deal with the singularity that appears in the rim of the disk by covering it by a circular torus.…”

Astigmatic Gaussian beams: exact solutions of the Helmholtz equation in free space

“…The most important example of a research of the first group is the theory of 'complex source' developed after pioneering papers [11,12] and [13] in both monochromatic and non-monochromatic cases. Singularities due to which these solutions do not satisfy the basic equation (1) in the whole space are described in detail in [14,15]. As an example of an exact solution satisfying (1) everywhere but involving a backward wave, we mention a nice simple solution found in [16].…”

Exact solution of Helmholtz equation for the case of non-paraxial Gaussian beams

“…Let us search for solutions of Eq. (1.1) in a classical form of Gaussian beams [7][8][9], which could be presented in Cartesian coordinate system X, Y, Z as below [10]:…”