Sinusoidal-Gaussian beams have recently been obtained as exact solutions of the paraxial wave equation for propagation in complex optical systems. Another useful set of beam solutions for Cartesian coordinate systems is based on Hermite-Gaussian functions. A generalization of these solution sets is developed here. The new solutions are referred to as Hermite-sinusoidal-Gaussian beams, because they are in the form of a product of Hermite polynomial functions of either complex or real argument times sinusoidal functions of complex argument times Gaussian functions of complex argument. These beams are valid for propagation through systems that can be represented in terms of complex beam matrices, and the previous beam solution sets are special cases of these more general results. Propagation characteristics and applications of these beams are discussed including their use as a basis set for propagation of arbitrary electromagnetic beams..

*Last Modified April 1998*