Task 2.1: Solution of electromagnetic simple problems on fractal domains

• Deliverables: D3 T2.1 Final Task Report
• Outputs: Closed-form results to be used as a consistency check for WP3 and WP1.

The application of the vector calculus developed in [9] and of the Green function renormalization techniques, applied for transport equations of scalar fields [10,11,12], to Electrodynamics which is intrinsically a vector problem needs some fundamental research to be developed in this workpackage:

The actions that will be done within this task are:

a) Connection between the topological formulation of field equations on simplicial complexes and the continuum limit (metric scale factor analysis).

b) Solution of paradigmatic examples of electromagnetism in the presence of fractal structures (e.g. the estimate of the inductance of a coil with fractal perimeter and the capacitance of a fractal surface; application of renormalization techniques to the analytical evaluation of dipole radiation patterns). The results obtained may be used in WP1 and in WP3 to check the computational methods developed.

c) Development of renormalization methods for vector-field equations in order to obtain the expression of the Green functions parametrized with respect to the order of iteration in the construction process of a fractal object. This action may be essential in approaching the quantification of the fundamental limits of fractal antennas (Task 1.3).

a) Connection between the topological formulation of field equations on simplicial complexes and the continuum limit.

The analysis electromagnetic fields on fractal supports leads to a topological formulation of the Maxwell equations. Since the domain in which the electromagnetic field propagate is fractal, the main issue is to reformulate the field equations in a way suitable to be applied for fractal lattices. This problem is extremely interesting for its physical and mathematical implications, but is rather aside the straight applications in engineering electromagnetic problems, especially as it regards antenna design. We think that this approach is not essential for the solution of antenna problems involving fractal structures. For that reason, the ROME reports corresponding to task 2.1 (Deliverable D3 and the Annex to this progress report) have reviewed the central ideas of this approach, without entering into the more technical details associated with the theory.

b) Solution of paradigmatic examples of electromagnetism in the presence of fractal structures

These problems involve the representation of sources radiating in free space and defined on fractal boundaries, which can be tackled by means of the formal apparatus developed in WP2 Task 2.2 (see Deliverable D4 and Annex to this progress report). This class of problems encompasses all the direct and inverse problems in antenna theory. The radiation problem of an antenna can be tackled by expressing the vector potential with respect to a current source distribution localized on a fractal support, the estimate of which can be obtained by solving the corresponding electromagnetic integral equation on a fractal support. The main issues are in this case the representation of scalar and vector fields on fractal structures, the setting of the integral equations on fractals and their solutions.

In the first place, direct electrostatic problems on fractals have been solved. By direct problems we mean that either the electric charge density or the electric current density are specified on a given structure. The classical integral equations involving charge/current density and scalar/vector potentials have been reformulated for fractal structures. The electrostatic potential of both a uniform and non-uniform parameterised charge distribution on a Koch curve and a non-uniform charge distribution within a Sierpinski gasket have been obtained.

In the second place, the solution of inverse electrostatic problem is addressed. Essentially, the inverse problem can be stated as finding the charge distribution on a fractal structure that produces a given scalar potential. It has been observed that, as the number of IFS iterations increases, the charge density –solution of the inverse problem- becomes a multifractal measure in the limit. Conversely, the charge measure –integral of charge density along the fractal curve- converges towards an invariant shape as the number of iterations increases.

The conclusions are:

• The charge density on a fractal structure displays a highly singular structure (resembling a multifractal measure). This is explained intuitively since fractal conductors display corners at all the lengthscales, and the singularity in the charge distribution reflects the presence of them.
• The solution of the inverse problem is well posed in terms of a charge measure rather than by enforcing charge densities. A limit charge measure, solution of the inverse electrostatic problem is numerically attained after few iterations in the construction process of the fractal structure.
• The highly singular structure of the charge measure prevents the use of collocation approaches for numerical discretization, since they make use of localized functions.

The behaviour of pre-fractal capacitors have been analysed as a function of the spacing between the two plates, and have been compared with the behaviour of parallel-plate condensers. Specifically, for the third iterate in the generation of the modified Koch curve, the fractal capacitor holds a much larger (factor larger than 8) overall charge than a Euclidean one, possessing the same overall end-to-end length.

In the third place, the fully vector radiation problem is addressed. The vector potential and the radiation pattern have been obtained for uniform and sinusoidal current distributions on modified Koch pre-fractal antennas after 3 to 5 IFS iterations. It is remarkable that the radiation pattern converges towards a limit just after a few iterations.