Fractal structures have been widely used in many fields of science, such as biology, physics and chemistry. In this article, we analyze the basic properties of a fractal-like square lattice of air holes, with most of the holes having a lattice constant of Λ while others are repeated with a lattice constant of 2Λ. We change the radii of these holes and analyze their effects on the bandgap regions and transmission properties. The analysis conducted here is based upon band diagrams and 2D Finite difference time-domain (FDTD) solution of the full-wave Maxwell's equations. We show that this structure provides flexibility in tuning the bandgap of the photonic crystal structure and we show the appearance of mini-bandgap regions along certain directions.
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