Two-dimensional fractal gratings with variable structure and their diffraction

Our interest is to treat the relationship between the geometry of the object and the corresponding intensity distribution in the scattered fields. The mathematical method for representing complex objects is important for the study of the fine structure in the Fraunhofer region. In the present work we make two considerations about the construction method that facilitates us the study of the scattered fields based on more simple structures, such as periodic distributions which were widely used in classical optics. The first consideration is an extension of a previous result on Cantor gratings for two-dimensional fractals with a variable fine structure. The mathematical foundation for these cases is related with the intersection between sets. The second consideration is the construction of other fractal sets, such as the Koch snowflake, through the union operation. In each case different aspects of the obtained diffraction pattern are shown. Also, the contribution from the simple components of the structures are taken into account for future applications in dynamic optical processing.