Complex point-source representation of real point sources and the Gaussian beam summation method

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Abstract
A new exact representation for point sources is given in terms of complex point sources. In the simplest configuration, a point source is equivalent to a distribution of sources on the surface of a sphere in complex space. The representation can be used to consider the propagation of point disturbances through inhomogeneous media and across interfaces. In the high-frequency limit, these solutions may be obtained by the use of complex ray-tracing methods, which are just the analytic extension of ordinary ray methods. It is shown that the additional use of the paraxial approximation yields a procedure that is similar to the Gaussian beam summation method. The latter technique is normally based on a matched asymptotics argument. However, the complex point-source representation now offers an exact basis for this widely used method.