Nowadays with the advance of the technology there are possibilities to fabricate promising new type of optical materials with dimensions comparable with the wavelength of the visible light. Photonic crystals are one type of these novel materials; they consist of periodic structures, usually fabricated from dielectrics.

The behaviour of electromagnetic waves in these structures can be understood with the help of Maxwell equations. At the base of all devices fabricated from photonic crystals stands a property which is a consequence of the periodicity: the dispersion relation shows allowed and forbidden bands.

Therefore to get a general understand of photonic crystals, in the first part of my talk I will investigate the mechanism behind the photonic band gap formation in a simple periodic structure which can be treated analytically.

The obtained results will be very similar to those describing the behaviour of electrons in real crystals suggesting a very general kinematic wave property. The Finite Difference Time Domain Method (FDTD) is a numerical technique to solve the Maxwell equations.I have developed several FDTD codes and I have applied them to compute different photonic structures [1]. Consequently in the second part of my talk I will cover briefly the main ideas of the FDTD algorithm.

Then I will present computations of photonic crystals which can be used as mirrors, resonators, waveguides and focusing lens elements. From the results of the FDTD simulations I produced several animations, which provides a visual picture of time and space evolution of electromagnetic fields and energy leading to a better understanding of the complex and interesting physical phenomena which take place in photonic crystals.