In our present work, we have made an investigation of several theoretical tools using the finite element method depending on the COMSOL MULTIPHYSICS program, for intuitive insight into the optical properties of the optical crystal. Analysis of the bandgap of a two-dimensional periodic photonic crystal with square lattice, evaluate the photonic band structure by the eigenfrequency of the unit cell of the first Brillouin zone. Moreover, creating defect mode inside the bandgap of photonic crystal, such as a resonant cavity, waveguide defect, narrowband filter, sharp drop filter, channel drop filter, waveguide bends, waveguide splitter promises us to a platform to design devices that includes a certain optical range of wavelengths. The transmission, as a result, the influence of the light localized in the defect area of the periodic structure of the photonic crystal. This study is important for producing photonic integrated circuits based on photonic crystals for future advanced optical communication.