Duality and similarity properties of the effective permittivity of two-dimensional heterogeneous medium with inclusion of fractal geometry

We report a systematic finite-difference time-domain study on the dielectric properties of two-dimensional lossless heterostructures with inclusion of deterministic fractal geometry (Koch's snowflake, Sierpinski's square and triangle) constrained to sit in a circumscribed circle, for a wide range of surface fractions. In all the configurations investigated, we observed (i) a strong deviation of the surface fraction dependence of the effective permittivity based on Maxwell Garnett analysis, and (ii) a permittivity change with reduced perimeter according to a similarity transformation, at least for the first three iterations of the fractal pattern. We show that the results of our numerical analysis disagree with expectations from the duality (phase interchange) relation, and explain this as being due to the intrinsic complexity of the morphology which requires a multipolar approach to correctly describe the microstructure features of these heterostructures.