Abstract In recent years, optical technology and photonics industry developed fast, but further progress became difficult due to a fundamental limit of light known as the diffraction limit. This limit could be overcomed by using the novel technology of nano-optics or nano-photonics in which the size of the electromagnetic field is decreased down to the nano-scale and is used as a carrier for signal transmission, processing, and fabrication. Among those systems we focus our interest on the nano-photonic crystals (PhCs) which are periodic, optical nanostructures represented by natural or artificial structures with periodic modulation of the refractive index. In this thesis, we present a quantitative study of the photonic crystals, in- one and - two dimensions, with higher symmetry (including the dispersion of the nano-systems). The theoretical treatment of PhCs could be done by different numerical techniques, using matlab program, as Plane wave expansion method, Finite difference time domain, Order-n spectral method…etc. In this work we examine one of the numerical methods for photonic crystal analysis, the FDTD. As application we study a design of dielectric Bragg mirror (DBM) to achieve high reflectivity. We used dielectric materials silicon dioxide (SiO2, known as fused silica) and titanium dioxide (TiO2) arranged as a periodic stacks to design Bragg mirrors using the silicon dioxide as a substrate. DBRs can be regarded as one dimensional photonic crystal with a high reflectivity stop band. We adopted the transfer matrix method to calculate the reflectivity spectrum of multi-layered dielectric films, with a controlled reflectivity and dispersion in the wavelength range 600 - 1400 nm, and showed it exhibits a reflectivity of > 99.99 % around 1000 nm.