Executive summary and project conclusions

Project objectives:

The project objectives, as stated in the Appendix I of the contract, are:

1. Increase the know-how in Fractal Electrodynamics theory and understand better the behavior of electromagnetic fields and electric currents in fractal domains, in order to acquire guidelines for the design of fractal-shaped antennas and microwave devices.
   
2. Explore if fractal-shaped microwave devices can reach the fundamental miniaturization limit, which has been never reached by Euclidean-shaped devices.
   
3. Develop a software tool for computer simulation of fractal-shaped microwave devices performance, including time domain visualization of the interaction between geometry and electromagnetic fields, in order to allow a physical interpretation of radiation and resonance of the proposed structures. This tool would allow also the later design and optimization of such devices.
   
4. Explore the impact of the technological limitations on the performance of fractal-shaped microwave devices, including minimum detail size and loss efficiency.
   
5. It must be remarked that this is a FET project and therefore the main objective is an increase of knowledge. Fractal antennas are becoming increasingly popular and many European SMEs, driven by the need of offering up-to-date state-of-the-art products, may decide to tackle risks by including them in their production lines. This project should provide answers about the potential interest of fractal antennas, through a careful study of electrical performance vs. technological complexity trade-offs.

The project has achieved all the five goals set up in the proposal, and additionally has paved the way to new technological applications.

The goals of the project were twofold. In one hand to develop the theoretical, and software tools to better understand fractal electrodynamics, and in particular the radiation and scattering of fractal structures, and on the other hand to design, build, and assess the properties of prototypes in order to validate theory and evaluate technological limitations.

In particular, the project has found what are the fundamental and technological limitations of miniature pre-fractal antennas and microwave devices, and how do they perform compared with the conventional ones. Small antennas design guidelines have been derived. Among the many miniature antennas and devices that have been designed, analyzed and/or measured, the ones that have an outstanding performance are the two-arm square spiral antenna and the Hilbert superconductor resonators for miniature filters. The project results are an important contribution to European SMEs interested in designing and manufacturing miniature antennas and microwave devices.

Main contributions:

The definition of pre-fractal geometries through an IFS allows to create extremely complex structures. Nevertheless, their intrinsic regularity simplifies their definition, modeling and somehow its numerical analysis.

• From the theoretical point of view it has been shown the well-posedeness of fractal electrodynamics and the convergence of classical EM numerical approaches applied to fractal domains. The main contributions in this field are:
  
  • Rigorous formulation of field problems on fractal structures (and specifically fractal curves).
    
  • Fully based on IFS through the concept of augmented IFS.
    
  • First rigorous theoretical formulation of EFIE on fractal wire antennas.
    
  • Numerical approaches based on Galerkin truncation of EFIE on wire antennas parametrized with respect to the normalized curvilinear abscissa.
    
• The study of the interaction of pre-fractal geometries with EM fields provides knowledge on properties of extremely complex structures otherwise difficult to attain.
  
• The conclusions derived from the behavior of small wire pre-fractal antennas lead to useful design criteria for better small wire antennas.
  
• The numerical analysis tools developed for the study of small wire pre-fractal antennas can be used with advantage in the analysis of complex high-detailed small near-resonance structures, such as PBG, metamaterials, etc. The time domain code developed during this project has been an invaluable tool for understanding the physical processes related to the behaviour of pre-fractal antennas. Specifically it has been very useful for understanding the role played by the “short cuts”.
  
• An intuitively user-prone approach to complex pre-fractal definition and automatic meshing has been developed and fully tested. This tool allows for the generation of thin wire as well as planar geometries. CIMNE will exploit all the utilities for the mesh generation of Fractal geometries by incorporating them in the commercial version of GiD. In this way, the construction of meshes for Fractal geometries is a new utility of the pre-process capabilities of GiD and, therefore, it has been made available to the scientific and engineering community, not restricted to the EM community. A free license of GiD will be provided by CIMNE to the rest of partners of the FRACTALCOMS project for, at least, two additional years after the end of the project.
  
• The knowledge gained in understanding the behavior of small pre-fractal wire antennas can be applied in using pre-fractal geometries to other EM devices.
  

Understanding fractal electrodynamics phenomena:

From the specific point of view of antenna miniaturization keeping reasonable bandwidth and efficiency, the following conclusions have been reached, based on the results of both numerical simulations and prototype measurements:

• Increasing the number of pre-fractal iterations means a reduction on resonant frequency, radiation efficiency and an increase on quality factor. The increase of fractal dimension, although making better space filling curves, builds larger monopoles with lower efficiencies and higher quality factors even for the first iterations.
  
• Topology has a stronger influence than fractal dimension on the behaviour of small 2D pre-fractal wire monopoles, in particular on the losses efficiency.
  
• As the number of loops inside the structure increases, efficiency and fractional bandwidth (inverse of quality factor) seem to increase with the order of the pre-fractal (number of IFS iterations).
  
• When there is no loop, each IFS iteration increases the length and bending of the wires, and as a consequence ohmic losses and the amount of stored energy on the surrounding of the antenna increases (this means lower radiation efficiencies and higher quality factors).
  
• When the number of IFS iterations increases beyond a certain threshold, the change in radiation patterns and input impedance of the antenna tend to zero. In other words, there is no use in increasing the number of IFS iterations. Convergence is usually achieved between 4 and 6 iterations. This value depends largely on the size, wire or strip width and topology of the antenna.
  
• The hypothesis of electromagnetic coupling –or shortcuts- between corners fully explains why the resonant frequency of pre-fractal antennas is much larger than what could be expected from the wire length only and why it stagnates as the number of IFS iterations increases.
  
• It seems that the high-gain localized modes than have been previously observed in the Koch-island printed patch antenna are not exclusive of pre-fractal antennas.
  
• Three-dimensional pre-fractal design does not provide further improvements than planar design in the radiation performance of monopoles. In spite of the smaller electrical sizes attainable thanks to their increased space-filling capability, they have a more intricate topology and larger wires than their planar counterparts. Consequently, efficiency and Q factor for these 3D pre-fractals have unpractical values to real-world applications.
  

Guidelines for the design of small antennas:

As a result of the work in this project (the electromagnetic coupling hypothesis) and very recent work available in the literature, some guidelines for the design of small antennas have been derived:

1. In order to reduce signal coupling –or shortcuts- between wire segment angles, the distance between those angles must be as large as possible, and the angles the larger possible.
   
2. In order to reduce the signal coupling between the feeder and the wire segments, the most possible wire length must be perpendicular to the electric field radiated by the feeder.
   
3. In order to reduce coupling between wire segments, parallel wire segments with opposite (anti-parallel) currents very close to each other must be avoided.

An example of wire antenna that closely follows these guidelines is a two-arm square spiral. The resonant frequency of a square spiral is inversely proportional to the wire length, while keeping the wire enclosed by a small square.

Fundamental limits:

The fundamental bandwidth limitation has been studied. It has been empirically assessed that the fundamental Chu-McLean limit of radiation quality factor of antennas holds even for pre-fractal devices. Moreover, pre-fractals are not closer to this borderline than other standard conventional designs.

• A multi-objective optimisation technique based on Genetic Algorithms (GA) assessed the existence of practical limits more restrictive than the fundamental limit predicted by Chu and reviewed by McLean. Both conventional and pre-fractal geometries are near this practical limit. The limit has been found for planar self-resonant wire monopoles.
  
• A more realistic limit that holds for antennas with sinusoidal current distribution has been recently published. The practical bandwidth limit found in this task agrees very well with the new theoretical limit recently found by Thiele et al. in 2003.
  

Technological limitations:

From the point of view of technological limitations two issues must be considered. First the limitations related to the modeling and numerical analysis of highly iterated pre-fractals, and second the limitation related to their manufacturing. Concerning these two issues the following conclusions have been reached:

• The best way to accurately model highly iterated pre-fractal curves is a extrusion-strip model rather than a planar strip or a thin wire. For that reason, most of the simulations in WP3 aimed at drawing conclusions for WP1 have been made with extrusion-strip models.
  
• Thin-wire models with reduced kernel are still useful for analysing low-order pre-fractals, and therefore NEC and the time-domain DOTIG code have been extensively used in this project.
  
• In order to analyze highly iterated pre-fractals with the thin-wire EFIE, a new formulation has been developed to achieve a very fast evaluation of the full kernel. The new formulation is valid for any ratio of the wire segment length to the wire radius, and can be applied to highly iterated pre-fractals discretized in very short wire segments. The main advantage of the new thin-wire full kernel approach over the 2-D extrusion-strip formulation is that computation time is very small, since the 1-D discretization requires, for instance, only 4096 unknowns while the 2-D mesh has 22,524 unknowns to model a 5-iteration Koch antenna.
  
• Technological manufacturing limitations make impossible the construction of highly iterated pre-fractals for reduced overall dimensions, due to the non-zero width of the pre-fractal curve. Nevertheless it has been shown, both theoretically and practically, that convergence is reached after a reduced number of iterations.
  

Design of pre-fractal devices:

Besides the objectives set on the proposal new approaches have been explored related to the following technological application of pre-fractals:

• Superconductor pre-fractal resonators for miniature filters.
  
• Quasi-self complementary pre-fractal antennas.
  
• GA optimized highly convoluted miniature antennas.
  
• H-trees and Capillary filters.