Computation of wave propagation in integrated optical devices - Koch, Thomas Bertold (ProQuest Dissertations & Theses, 1990)
This thesis describes a theoretical investigation using numerical methods to analyse the propagation of electromagnetic waves in integrated optical devices of arbitrary longitudinal and transverse geometry. The classical BPM (Beam Propagation Method) with its spectral propagation characteristic, is based on the assumption of scalar propagation for TE waves, low refractive index change and the neglection of reflected waves and has been found inadequate to model integrated optical devices that involve abrupt large changes of the refractive index in the transverse and longitudinal direction. In this work it is shown that an alternative formation of the propagation problem in terms of a x-transient variational formulation for the TB and TM Helmholtz equation overcomes these basic limitations of the classical BPM. The variational principle, formulated with initial and boundary conditions, is solved using a numerical method based on a combination of Finite Elements and Finite Differences. Solutions in anisotropic material are obtained by applying standard Galerkin Finite Element (FE) and Finite Difference (FD) methods to a weak formulation derived from the coupled TE/TM Helmholtz equations. Also, to counter the disadvantages of the formulation of the propagation problem in terms of an initial boundary value problem with its inherent neglection of backward travelling waves, a hybrid method is developed to include possible reflected waves at each propagation step. To obtain the field solution in the optical chip including all reflected fields, the partial reflections are superimposed and subsequently propagated several times in the forward and backward direction using the FE/FD algorithms instead of the BPM. A series of numerical simulations are presented which provide vector and scalar solutions of the TE and TM beam propagation and reflection in linear and nonlinear waveguides involving large transverse changes in refractive index. Straight, curved and Y-shaped waveguides as well as the reflection of Gaussian beams, TE and TM modes at single and multiple longitudinal discontinuities have been analysed. An error analysis shows that the new algorithms are unconditionally stable and faster than the BPM. These results reinforce present theory and demonstrate the validity of the new algorithms developed.