In this paper, we have studied the photonic band structure of function photonic crystals in which the dielectric constant of the scattering centers (rods) is a function of space coordinates. The under-studied lattice is hexagonal and cross section of rods has a circular symmetry embedded in the air background. Photonic band structures for both electric and magnetic polarizations of the electromagnetic waves are calculated. The obtained results show the existence of the forbidden frequency region (photonic band gap). It is considered that the dielectric rods are made of the Kerr type materials. Therefore, by considering different distributions of light intensity, different function forms will be obtained for the dielectric constants of rods, which are called function photonic crystals. The influence of the function coefficient (corresponding to the Kerr coefficient) on photonic band structures has been theoretically investigated. The results show that the width and number of photonic band gaps are more controllable than the conventional photonic crystals. These results can be very useful in designing the optical devices.


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