‘Complex source’ wavefields: sources in real space

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*.

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