We show that a 3-D microwave crossing waveguide can be solved by a 2-D scalar Helmholtz equation with combining boundary conditions for TE and TM modes. Furthermore the crossing waveguide possesses a symmetry along two diagonal axes passing through the origin. Computation of the EM wave fields is decomposed into four smaller tasks of computing reflection coefficient vector of a parallel plate waveguide terminated with a corner made of two perfectly electric or magnetic conducting walls (PECW/PMCW). In this thesis, we propose a mixed Cartesian and polar coordinate mode-matching method to solve this 2-D corner cube microwave reflection problem. The solution is obtained by applying the continuity condition of both the tangential field and its normal derivative along a given curve inside the overlapped region of the two coordinate systems. We are able to compute up to the third decimal place of the reflection, through and cross transmission coefficients. All results pass the energy conservation test and are verified and compared with those computed by Integral equation method simulation.

Dielectric waveguides are crucial devices in the making of integrated-optical circuits. It is very important to analyze this type of waveguides so we can optimize the design for better performance. Analysis of bent waveguides has been a difficult problem in the past. In a bent waveguide, two coordinate systems are needed to fully describe the ongoing complex scattering process in the transition region of the waveguide. It is extremely hard to analyze such problems for methods built on a single coordinate system such as the finite-difference,finite-element methods and the beam propagation method (BPM). In this thesis, we adopt dual mode-field representations (for all the low and higher-order modes), one for the incident and reflected waves and the other for the transmitted waves, to study bending effects. To calculate the wave fields, we apply the analytic continuity principle to allow the waves to analytically extend and join smoothly on the bordering line. By matching the two continuity conditions of both the fields and their normal derivatives we get two matrix equations for the reflection and transmission coefficients. For symmetrical bending waveguide, the task can be further reduced to solving two smaller problems each with even or odd symmetry on the bordering line. As the bent angle increases the governing matrix equation becomes more singular. As a result, all the elements in the matrix are calculated with closed-form formulae to minimize the stability problem. In addition, special numerical methods are used to extend the range of the bending angles that this method can handle. In conclusion, our theory can calculate microwave bending waveguides up to 30 degrees and for dielectric slab waveguide with 15 degree bent angle. With this method we are able to compute small reflection coefficients of about -60dB and less.