Abstract
The invention of laser in 1970 has steadily increased the carrier frequency from the millimeter wavelength to micrometer range. Optical fiber communication systems using laser diode as the a light source and single-mode fiber as transmission media based on wavelength-division multiplexed (WDM) technology with the transmission capacity of 3.2 Tb/s or more are available now. With the rapid growth of semiconductor manufacturing technology, considerable efforts have been directed to computing the modes of optical rib waveguides, which form the important parts of photonic integrated circuits. In this dissertation, we study the solutions of waveguide problems by novel numerical methods on two waveguide structures. The first one is a multimode interference coupler with longitudinally invariant structure. In this structure, we propose a method to study wave propagation in longitudinally invariant waveguides with arbitrary index profile. According to our method, both the electric field and the refractive index profile are expanded into two Fourier cosine series. With these series substituted into the wave equation, a differential matrix equation can then be obtained. We show that such a matrix equation can be explicitly solved and an expression for the wave field at any longitudinal position along an optical waveguide can be obtained. The solution proposed in this method indicates that our approach yields the same results as those obtained by using the beam propagation method with approximation. The second novel numerical method is on solving rib-type waveguide problems. A new semi-analytic method for solving the modal indices of the optical rib-type waveguide problems is presented. In this method, the cross-section of a rib-type waveguide is divided into several regions. In each region, the refractive index profile and field distribution are expanded into Fourier cosine series, and then are substituted in the wave equation. A second-order differential matrix equation is then derived for each region, and a closed-form solution can be obtained. Given the boundary conditions, an eigenmode equation for the rib waveguide can be derived and solved numerically to give the modal indices. The method proposed here is used to deal with three rib waveguides in three different geometric dimensions and/or compositions, respectively. Computational results indicate that our method is quite efficient, in terms of CPU time and its accuracy. The relative error in computing the modal index with the method is about 10-5 ~10-6. In addition, a new design for beam splitting components employing silicon-on-insulator rib waveguide structures is presented. In this design, a high index thin film layer is deposited in the rib section to reduce the wave field dispersive tails in the slab section. Accordingly, it renders the mode field a confined spot. In terms of the excess loss, fiber coupling loss and compactness of these components, this structure improves the beam splitting performance as compared with some conventional waveguide components such as branches and multimode interference couplers (MMICs).