In this thesis, we demonstrated the effectiveness and the accuracy of the FD-FD method for complex optical waveguide structures such as the micro ring resonator, micro disk resonator, tapered waveguides and waveguides terminated with tilted facets. We are able to achieve the goals by deriving the following modification/extension of the original FD-FD methods. In frequency domain, we can build an accurate frequency-domain modal absorbing boundary condition (ABC) for both the homogeneous region and for the layered background. This allows us to connect the analytical modal solutions with FD solutions and thus reduce the area of the FD domain. In addition, we adopt an effective index averaging method for representing equivalent material for grid cells containing more than one kind of materials. For the TM case, for each grid cell we need to compute effective indices for all four surrounding cells (left, right, up, and down). For the TE case, we need to compute just one effective index within each grid cell. Note that we employ two different averaging schemes for the TE and the TM cases. To solve the huge block tri-diagonal matrix equation (derived from the FD-FD approximation) we modified the Thomas method and we were able to obtain the solutions of linear equations involving more than a hundred thousand variables under a few minutes. We used our method to analyze optical micro-ring waveguides, micro-disk cavities, adiabatic tapered waveguides and waveguides terminated with tilted facets. The simulated results include the reflection coefficients, transmission coefficients and field distribution.

Radio wave absorbing materials (RAM) are commonly found amongst high-tech products such as LCD electronic devices, laptop and desktop computers. Electromagnetic wave absorbing materials are composed of dielectric materials mixed with ferrite, a magnetic material, with varying shapes and sizes. It should be capable of absorbing electromagnetic energy at normal and large incident angles over a wide range of frequencies. This requires the material to possess a large relative complex dielectric constant (permitivity £`r), as well as a large relative complex magnetic permeability constant (£gr). Due to the nature of the complexity of the RAM, which surpasses standard analysis techniques, we have derived, for this thesis, frequency-domain two-dimensional finite-difference formulas for modeling the electromagnetic behavior of RAM. This involves using a material that has a given £`r(1:10 range) and £gr(1:1000 range) which covers a vast range of indices of refraction. To reduce the computational domain, we took care of implementing the numerical absorbing boundary conditions, while also implementing material averaging schemes for the finite-difference coefficients that cover the region where sample medium changes. Simple numerical examples are included to verify our mathematical model. We also implemented an optimal one-dimensional multi-layered RAM design, designed by using a constrained optimization searching technique. Included in the thesis are two complete, practical, optimal designs considering available material parameters (finite loss tangent) as well as their actual manufacturing limitations (layer thickness).