Complex-source-point theory of the two dimensional hermite-gaussian beams

It is shown that, by assigning complex values to the source co-ordinates of an oscillating multipole line source in free space, a highly collimated field can be generated that behaves paraxially like a two-dimensional Hermite-Gaussian beam. The result can be used for the asymptotic treatment on high frequency propagation and scattering. Moreover, a new mode designations scheme for the beam modes is proposed.