Multiple dielectric crossing waveguides are indispensable in building a complex optical integrated circuit. Since each input/output waveguide will have many crossings, it is important to design a low-loss waveguide crossing to ensure the overall radiation loss is kept at a minimum. The beam propagation method (BPM) is usually the method of choice for modeling large but low-index-contrast waveguide devices. BPM assumes one-way propagation and adopts the paraxial approximation. It is neither able to consider reflection of electromagnetic (EM) fields nor to perform wide angle propagation of forward fields. Therefore, it can not be used to analyze perpendicular dielectric crossing waveguides. At a maximum 0.5 dB power loss per crossing, the difficulty of simulation a waveguide crossing is how to compute the complex coupling waves with high enough precision. In this thesis, two-dimensional planar integrated optical waveguide crossing is studied in detail for the through and cross power coupling coefficients with the finite-difference frequency-domain (FD-FD) method. By exploiting the dual symmetries: the ¡§+¡¨ symmetry and the ¡§X¡¨ symmetry in the perpendicular crossing waveguide, we are able to compute the EM fields and their power coefficients without using artificial absorbing boundary conditions (ABC) nor using the perfectly matching layer (PML). We develop the layer-mode based transparent boundary condition (LM-TBC) [1] for launching the fundamental incident mode as well as transmitting the reflected and scattered wave fields off the crossing area. Numerical results including the field distribution, power coefficients are carefully verified and the convergent comparisons are also studied in the thesis.