Efficient marching solver for the three-dimensional Helmholtz equation in the cuboid waveguide

In this paper, a marching solver designed for the computation of wave propagation is developed within a class of open three-dimensional waveguides. For the wave propagation model, a three-dimensional scalar and frequency-domain Helmholtz equation is effectively adopted. Specifically, the boundary value problem is converted into an initial value problem by the Dirichlet-to-Neumann (DtN) mapping. Then, with the solving domain restricted to a class of cuboid waveguides, a numerical marching method based on DtN mapping is well established. Besides, when a class of open waveguides is included in the solving domain, the perfectly matched layer (PML) can be used to change the open domain into the bounded form (cuboid waveguide). Finally, the original Helmholtz equation now is equivalent to a complex differential equation with the second order, whose corresponding numerical marching method is also triggered. Numerical comparisons show that the marching method based on the DtN mapping is more efficient and feasible than the one-way method (that is, the beam propagation method (BPM)) when solving the three-dimensional Helmholtz equation.