A beam propagation method based on the Galerkin method with Hermite-Gauss basis functions for studying optical field propagation in weakly guiding dielectric structures is described. The selected basis functions naturally satisfy the required boundary conditions at infinity so that the boundary truncation is avoided. The paraxial propagation equation is converted into a set of first-order ordinary differential equations, which are solved by means of standard numerical library routines. Besides, the calculation is efficient due to its small resulted matrix. The evolution of the injected field and its normalized power along the propagation distance in an asymmetric slab waveguide and directional coupler are presented, and the solutions are good agreement with those obtained by finite difference BPM, which tests the validity of the present approach.
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