Towards a Theory of Electrodynamics on Fractals:representation of Fractal Curvesand Integral Electromagnetic Equations in Fractal Domainsabstract

Fractal shaped antennas and fractal miniaturized elements are becoming a promising new family of devices with a wealth of potentially useful applications [1]. Owing to the facility of manufacturing fractal devices (the metallisation process of fractal antennas up to a few iterations of the recursive process that generates the fractal structure is a relatively simple task), several different prototypical fractal antennas have been produced and analysed. The experimental and computational researches on such structures still lack a solid theoretical counterpart, which should yield a comprehensive framework for the definition and the analysis of vector fields defined on fractal subsets. As a consequence of this, some fundamental issues of electromagnetic theory for fractal radiators are still unsolved (e.g., whether fractal dipolar radiators are subject to the same fundamental limits of Euclidean ones). Existing literature cannot be of help for solving these problems, since it is intrinsically limited to structures which are the initial iterations in the construction process of fractal structures (prefractals) [2,3]. The aim of this work is to propose a systematic theoretical analysis of direct and inverse electromagnetic problems on fractals. We limit here to fractal structures which are topologically equivalent to one-dimensional curves (fractal wires). Two main issues are analysed: i) parametric representation theory of fractal structures by means of Augmented Iterated Function Systems; ii) definition of vector fields on fractal domains and solution of inverse problems expressed as integral (scalar and vector) equations defined on fractal supports. It is important to stress out that the definition of vector fields on fractal structures involves delicate theoretical issues, related to the definition of tangency properties on fractals. This can be intuitively argued by considering tangential current distributions on wire antennas defined for structures, like fractal curves, for which tangent vectors cannot be defined almost anywhere in a classical way, and the definition of which requires nontrivial concepts of measure theory.