High-Precision And Fast Computation Of Wave Propagation In Open Waveguide

For acoustic, electro-magnetic, and seismic wave propagation problems of practical interest, it is important to solve the wave propagation with high-precision quickly. The length scale along the waveguide is typically very large. The transverse length scale is much smaller, but still much larger than the char-acteristic wavelength. Standard numerical techniques such as the finite difference and finite element methods lead to a system of equations with a very large number of unknowns and are not very practical for these large-scale problems.For the bounded waveguides along the transverse direction, many researchers had studied the computation of the wave propagation in the gradient refractive-index (GRIN) optical waveguides or in the acoustical waveguides with varied wavenumber. There are some efficient computational methods for the wave propagation, such as the beam propagation method for GRIN slab waveguides or for complex systems, the operator formalism for paraxial propagation in ho-mogeneous and inhomogeneous media,and the marching scheme based on the Dirichlet-to-Neumann (DtN) map for weakly range-dependent waveguides.For the unbounded waveguides along the transverse direction, by using the perfectly matched layers (PMLs) the transverse plane can be truncated to a bounded and relatively small region. Therefore, the propagation problem is for-mulated in a domain with just one direction (i.e. transverse direction) having a particularly large length. For this special geometric feature, the operator march-ing method (OMM), namely an one-way re-formulation based on the DtN map, can be used to compute the wave propagtion efficiency.In OMM, there is a local base transformation to be done in each marching step by searching a coordinate matrix related with the eigenvalue problem. For a bounded waveguide, the transverse operator is self-adjoint, and its system of the linearly independent eigenfunctions forms an orthogonal basis. Thus, in this case, the process of the local base transformation is easy to do. However, for an unbounded waveguide, a PML is applied to terminate it to a bounded region, and the original Helmholtz equation with real coefficients is transformed into a partial differential equation (PDE) with complex coefficients. So, the resulting transverse operator of the PDE is not self-adjoint. It leads to lose the orthogonal-ity for the eigenfunctions’system. This brings great difficulty in the numerical implementation of the local base transformation. The adjoint operator method (AOM), which is constructed for lossy waveguides, can be used to implement the local base transformation. However, our research shows that an unacceptable error will be produced if we implement the local base transformation by AOM.In this thesis, a new treatment for the local base transformation is provided by studying the transverse operator. We also investigate the relationship between the two methods through theoretical derivation. It is shown that the new treat-ment is equivalent to the AOM if the transverse operator is discretized on infinite grids. But in terms of practical effect, the new treatment is more efficient than the AOM. Some high-precision results of optical propagation in GRIN media are obtainedWe also study the wave propagation in open waveguide with a curved in-terface. Firstly, we use a local orthogonal transformation to flatten the curved interface. Then, combined with the new treatment of the local base transforma-tion, OMM is used to solve the Helmholtz equation efficiently.