“Complex source” in 2D real space

The paper concerns the complexified Green function $g_*$ for the 2D Helmholtz equation, which is studied as a non-paraxial model of a Gaussian beam. The function $g_*$ satisfies a certain inhomogeneous Helmholtz equation in the real space with source distribution depending on the choice of the branch of certain complex square root. We discuss various choices of the branch cut and calculate the corresponding source distribution.