- 1 -

FRACTALCOMS Exploring the limits of Fractal Electrodynamics for the future telecommunication technologies

IST-2001-33055

Deliverable reference: D10

Contractual Date of Delivery to the EC: January 31, 2004

Author(s): José M. González and Jordi Romeu

Participant(s): UPC and EPFL

Workpackage: WP4

Security: Public

Nature: Deliverable

Version: 1.0

Date: 15-12-2003

Total number of pages: 15

Keyword list: antenna, fractal, fabrication, Koch, Hilbert, Sierpinski

Task 4.2 Final Report Technological Limitations

of Fractal Devices

Abstract:

Practical limitations on the fabrication of fractal devices are considered. Examples of these limitations are shown for the most common geometries.

- 2 -

Table of Contents

1 INTRODUCTION.................................................................................................... 3

2 LIMITATIONS IN THE MANUFACTURING PROCEDURES ........................... 5

2.1 Koch curves: technological limits .................................................................... 5

2.2 Hilbert curves: technological limits.................................................................. 8

2.3 Sierpinski curves: technological limits........................................................... 11

3 LIMITATIONS IN THE MANUFACTURING PROCEDURES at UPC............. 12

4 CONCLUSIONS .................................................................................................... 13

5 REFERENCES ....................................................................................................... 14

DISCLAIMER................................................................................................................ 15

- 3 -

TECHNOLOGICAL LIMITATIONS OF FRACTAL DEVICES

1 INTRODUCTION

Fractal is, according to the definition given by B. Mandelbrot [Mandelbrot, 1977], a set with a Haussdorf-Bessicovitch dimension strictly higher that its topological dimension. This definition covers several geometries that are self-similar under several scales [Peitgen, 1992]. Some of them have been carefully analysed under different scopes in the field of Fractal Electrodynamics. For instance, the Koch, Minkowski, Hilbert, and Peano curves; the Sierpinski gasket; the Sierpinski carpet and the Koch Island are typical examples that could be found in the literature of Antennas. Figure 1 shows these designs.

All of these intricated geometries are built using an iterative algorithm called Iterative Function Systems (IFS), which can be expressed mathematically by (1)

[ ] ∞== − ,...,1,0 1 iAWA ii (1)

where W is the Hutchinson’s geometrical transformation algorithm

[ ] ( ) ( ) ( )AwAwAwAW N∪∪∪= ...21 (2) being w an affine transformation.

Fig. 1 Typical designs of fractal geometries.

- 4 -

After an infinite number of iterations, the initial set A0, or initiator, which started the iterative algorithm, is transformed into the set A∞ that is the fractal. Each intermediate state Ai is called pre-fractal. A graphical example of how an IFS works is through the use the Multiple Reduction Copy Machine (MRCM). This copy machine has N lenses that simultaneously generate N copies of the object that is its input. The output from the first copy is the input for the second copy. This procedure goes on to infinite. Figure 2 shows how the MRCM method is used to generate a Sierpinski gasket. In this example, the copy machine has three lenses that scale down by a factor of 2 the input object. Both, MRCM and IFS, are connected. Each lens in the MRCM is an affine transformation in the IFS, the set of lenses in the MRCM is the Hutchinson’s geometrical transformation in the IFS or the generator in the fractal terminology, the input object to the copy machine at the beginning of the iterative algorithm is the initiator of the algorithm, and each output of the copy machine is a pre-fractal. These procedures for generating fractals, MRCM and IFS, reveal the first limitation that we have when working with fractal devices. They actually are the result of infinite iterative algorithms. So, from a practical point of view, we always work with truncated versions of these algorithms: we always work with pre-fractals.

Luckily, from an electromagnetic-wave point of view this limitation is not outstanding: from a certain iteration the electromagnetic waves are unable to resolve much smaller intricacies than a wavelength, and using highly iterated devices is not useful due to the high ohmic losses and the large stored energy in the surroundings of the pre-fractral (typically from iterations higher than 4, 5 or 6) [Gianvittorio, 2002].

Fig. 2 The Multiple Reduction Copy Machine.

- 5 -

However, if highly iterated devices would be needed, limitations in the manufacturing procedures and probably in the simulation of these structures would appear. For the vast majority of applications the large numbers of unknowns we have to deal with pose a serious constrain in the simulator we have to use. With a proprietary code it is easy to choose and implement a technique that face this problem and shifts the bottleneck of the design to the fabrication procedure. IFS is the procedure used to design pre-fractals for antenna and filtering applications under the frame of the project Fractalcoms.

2 LIMITATIONS IN THE MANUFACTURING PROCEDURES

When manufacturing a pre-fractal device the width of the curve is the value that makes the device attainable (manufacturable) or not. We do not mind if we are manufacturing these curves with wires or strips, or even if we are dealing with antennas, resonators or other devices. The dimension w will always refer to the strip width if we are designing microstrip devices, or a wire diameter if we were designing wired devices. In any case, we will refer to planar designs, i.e. fractal curves that are included in a plane, the most widely used in the Fractal Electrodynamics literature. In the next paragraphs the resolution of our tools or manufacturing processes to make strips or wires will stand the maximum iteration achievable for a given fractal design. This maximum iteration will not be the same if we are working with Koch, Hilbert or Sierpinski pre-fractals, or any other pre-fractal curve.

2.1 Koch curves: technological limits The technological limitation for the maximum iteration when manufacturing a Koch pre-fractal is connected with the width of the curve. We have stablished the limit between the manufacturability and non-manufacturability region for a given pre-fractal iteration when the width of the curve “fills the gap” of the equilateral triangle side of finest resolution. Figure 3 shows the geometrical arrangement of the variables that take part in the development that follows.

- 6 -

According to the previous paragraph and with figure 3, the criterion used to decide if a pre-fractal can be resolved or not is,

2ld ≤ (1)

where l is the length of a pre-fractal segment, and d is the slant half-width of the Koch curve. Applying to (1) the geometrical constraints of figure 3 we have

lw 3≤ (2) where the criterion is expressed in terms of the Koch curve width w and the length of each segment l. When the width w of the curve is large the zigzags are unresolved, as figure 4 shows for a Koch curve of first iteration and for several ratios w/l. Taking into account that the length of the segments is related with the pre-fractal iteration N and the size S of the curve by

NSl

3= (3)

we finally get equation (4) that summarizes the relationship between the maximum iteration N (an integer number) of a Koch curve pre-fractal that can be manufactured given a curve width w and curve size S:

−≤3log 3

logS

w

N (4)

In equation (4) the brackets indicate the floor of the argument.

Fig. 3 Geometrical arrangement of variables for the finest intricacies of a given iteration of a Koch curve pre-fractal.

- 7 -

Figure 5 shows the maximum achievable iteration for a Koch curve versus the w/S ratio. The value w will be given by the resolution of the manufacturing technique, and S by the size of the pre-fractal (designed to fulfil certain performances).

Fig. 4 Red lines show the Koch curve pre-fractal of first iteration for several values of the ratio w/l. While (a) and (b) are valid pre-fractal designs, (c) is a design where the pre-fractal intricacies are not resolved and the design is not attainable. The blue line represents the ideal pre-fractal curve with zero width.

Fig. 5 Maximum iteration of a manufacturable Koch curve pre-fractal with a given w/S ratio. The region over the curve includes the designs that cannot be fabricated. Red and blue dots show, respectively, the highest iterations simulated and manufactured for monopole antenna designs taken from literature.

- 8 -

The area over the curve represents the values that cannot be manufactured with our present technology (i.e. for a given w/S ratio). The area under and on the curve includes the iterations manufacturable with a given w/S ratio. Blue dots are typical w/S values of manufactured structures taken from literature ([Puente-Baliarda, 2000], [Best, 2003, c], [González, 2002]), while red dots are typical w/S values of simulated structures also taken from literature. All of these ratios correspond to monopole antenna designs. We should emphasize from figure 5 that the highest pre-fractal iteration manufactured is the 5th and that all of these designs referred in literature correspond to ratios w/S around 10-3 and 10-2. In figure 6 we show an example that clarifies the graph of figure 5. If the ratio w/S (ratio between the resolution w of our manufacturing technique and the size S of the curve) is 10-1, we have resolution enough to discern the intricacies of the Koch curve until the second iteration (a). For iterations higher than the second the pre-fractal will not be resolved (b).

2.2 Hilbert curves: technological limits The same procedure as in the previous section can be applied to other typical pre-fractal curves, as for instance Hilbert curves, used in Fractal Electrodynamics. For these curves the maximum iteration attainable with standard manufacturing techniques will also depend on the ratio w/S. In this case, S is the side length of the square where the Hilbert curve is enclosed in, and w is the width of the strip or wire that we are going to fabricate the curve with.

Fig. 6 Koch curve pre-fractal designs with values w/S=10-1: (a) iteration N=2; (b) iteration N=3. (a) Is considered valid, while (b) is a design where the intricacies of the wire/strip (red lines) do not have enough resolution to follow the zigzags of the ideal pre-fractal (blue line).

- 9 -

Equation (5) relates the segment lengths l of a given pre-fractal iteration of a Hilbert curve enclosed in a square of side length S

NSl

2= (5)

For this geometry the criterion to decide if the pre-fractal can be resolved or not is

lw ≤ (6)

as figure 7 shows for the smallest intricacies of a given iteration of a Hilbert pre-fractal.

Figure 8 shows three designs of a Hilbert pre-fractal of 3rd iteration manufactured with different values of the ratio w/l to graphically assess the criterion of equation (6). From equations (5) and (6) the ratio between maximum manufacturable iteration versus the technological capability of our fabrication processes w/S is obtained

−≤2log

logSw

N (7)

The maximum manufacturable Hilbert pre-fractal iteration for a given w/S is displayed in figure 9. Dots show the Hilbert pre-fractals of highest iterations (simulated and manufactured) whose dimensions are taken from literature on Antennas ([Anguera, 2002] [Best, 2002] [Best, 2003, a] [Best, 2003, b] [González, 2002] [González-Arbesú, 2003, a] [González-Arbesú, 2003, b] [Vinoy, 2001] [Zhu, 2003]).

Fig. 7 Details of the smallest intricacies of a Hilbert curve pre-fractal iteration. The variables w and l are, respectively, the curve width and the segment length.

- 10 -

Fig. 9 Maximum iteration of a manufacturable Hilbert curve pre-fractal with a given w/S ratio. The region over the curve includes the designs that cannot be fabricated. Red and blue dots show, respectively, the highest iterations simulated and manufactured for monopole antennas. These values are taken from literature.

Fig. 8 Red lines show the Hilbert curve pre-fractal of 3rd iteration for several values of the ratio w/l. While (a) and (b) are valid pre-fractal designs, (c) is a design where the pre-fractal intricacies are not resolved and the design is regarded not valid. The blue line shows the ideal pre-fractal curve with zero width.

- 11 -

2.3 Sierpinski curves: technological limits For a Sierpinski geometry, the criterion used to decide if the pre-fractal intricacies are resolved or not is

3lw ≤ (8)

being w the width of the manufactured pre-fractal and l the length of each

segment. Figure 10 shows, according to this criterion, two manufacturable designs of a Sierpinski pre-fractal and a design considered not attainable.

Accounting for the ratio between the side length S of the Sierpinski triangle and the length l of the segments

NSl

2= (9)

the maximum manufacturable Sierpinski pre-fractal iteration N versus the

technological resolution w/S is shown in figure 11. The highest iterations (simulated and measured) found in literature ([González, 2002]) are also displayed in the figure. The graph in the figure corresponds to equation

−≤2log

3logS

w

N (10)

Fig. 10 Red lines show a Sierpinski curve pre-fractal of 3rd iteration for several values of the ratio w/l. While (a) and (b) are valid pre-fractal designs, (c) is a design where the pre-fractal intricacies are not resolved and the design is not valid. The blue line shows the ideal pre-fractal curve with zero width.

- 12 -

3 LIMITATIONS IN THE MANUFACTURING PROCEDURES AT UPC

In the previous sections we have assessed through examples (figures 4, 6, 8 and 10) how the resolution in our manufacturing process affects the maximum iteration attainable for a given pre-fractal curve. Also we have shown figures 5, 9 and 11, that are very useful to decide if our designs would be manufacturable given a certain resolution (w/S) in our fabrication procedures. At UPC the standard procedures for manufacturing printed circuit boards allow the fabrication of strips as small as 100 µm on CuClad substrate (etching, 35 µm), 80 µm on Rogers substrate (etching, 17.5 µm), or 20 µm on a superconductor (etching, 200 µm). These figures together with the curves of figure 5 reveal that in the case of a ,for instance, Koch pre-fractal monopole with a resonant frequency around 1-2 GHz the maximum attainable iteration would be the 6th. If we were intended to fabricate a 10th iteration Koch pre-fractal monopole with a medium resolution technique of 80 µm, we would require a monopole of 8 m height, that would resonate at a frequency lower than 9.4 MHz. In both examples the frequency shift due to the substrate has been ignored (though in a real design they should have been taken into account).

Fig. 11 Maximum iteration of a manufacturable Sierpinski curve pre-fractal with a given w/S ratio. The region over the curve includes the designs that cannot be fabricated. Red and blue circles show, respectively, the highest iterations found in literature that have been simulated and manufactured.

- 13 -

4 CONCLUSIONS

Expressions that show the relationship between the maximum manufacturable iteration for a pre-fractal curve versus the resolution of our fabrication process has been determined for standard pre-fractals such as Koch, Hilbert and Sierpinski curves. These expressions can be used for any kind of fractal device using these curves. Expressions for other fractal curves should be determined following the method developed in this document. From figures 5, 9 and 11, and for monopole antennas, iterations higher than 6 are unattainable with standard manufacturing techniques for frequencies around 1-2 GHz. Higher resonant frequencies will quickly reduce the highest iteration achievable.

- 14 -

5 REFERENCES

[Anguera, 2002] J. Anguera, C. Puente, and J. Soler, “Miniature monopole antenna based

on the fractal Hilbert curve”, Proc. IEEE Int. Antennas and Propagat. Symp., 2002, vol. 4, pp. 546 –549.

[Best, 2002] S. R. Best, “A comparison of the performance properties of the Hilbert curve fractal and meander line monopole antennas”, Microwave Optical Technology Letters, vol. 35, no. 4, pp. 258-262, 20th November 2002.

[Best, 2003,a] S. R. Best, “A comparison of the resonant properties of small space-filling fractal antennas“, IEEE Antennas and Wireless Propagat. Let. , vol. 2, no. 13, 2003, pp. 197–200.

[Best, 2003,b] S. R. Best, and J. D. Morrow, “On the significance of current vector alignment in establishing the resonant frequency of small space-filling wire antennas“, IEEE Antennas and Wireless Propagat. Let. , vol. 2, no. 13, 2003, pp. 201–204.

[Best, 2003,c] S. R. Best, “On the performance properties of the Koch fractal and other bent wire monopoles”, IEEE Antennas and Propagation Magazine, 2003, vol. 51, no. 6, pp. 1292–1300.

[Gianvittorio, 2002] J. P. Gianvittorio, and Y. Rahmat-Samii, “Fractal Antennas: a novel antenna miniaturization tecnique, and applications”, IEEE Antennas and Propagation Magazine, Feb. 2002, vol. 44, no. 1, pp. 20-36.

[González, 2002] J.M. González, and J. Romeu, “On the Influence of Fractal Dimension and Topology on Radiation Efficiency and Quality Factor of Self-Rsonant Prefractal Wire Monopoles: Modelling with NEC and Preliminary Measurements”, UPC, Barcelona, Spain, European Comission, Fractalcoms Project (IST 2001-33055), Internal Report , Dec. 2002.

[González-Arbesú, 2003,a]

J. M. González-Arbesú, S. Blanch, and J. Romeu, “The Hilbert curve as a small self-resonant monopole from practical point of view“, Microwave and Optical Technology Letters, vol. 39, no. 1, pp. 45–49, 5th Aug. 2003.

[González-Arbesú, 2003,b]

J. M. González-Arbesú, S. Blanch, and J. Romeu, “Are space-filling curves efficient small antennas?“, IEEE Antennas and Wireless Propagat. Let. , vol. 2, no. 10, 2003, pp. 147–150.

[Mandelbrot, 1977] B. Mandelbrot, The Fractal Geometry of Nature, New York, W.H. Freeman and Company, 1977.

[Peitgen, 1992] H.O. Peitgen, H. Jürgens, and D. Saupe, Chaos and Fractals: New Frontiers of Science, Springer-Verlag, New York, 1992.

[Puente-Baliarda, 2000]

C. Puente-Baliarda, J. Romeu, and A. Cardama, “The Koch monopole: a small fractal antenna”, IEEE Transactions on Antennas and Propagation, November 2000, vol. 48, no. 11, pp. 1773–1781.

[Vinoy, 2001] K. J. Vinoy, K. A. Jose, V. K. Varadan, and V. V. Varadan, “Resonant frequency of Hilbert curve fractal antennas”, Proc. IEEE Int. Antennas and Propagat. Symp., 2001, vol. 3, pp. 648 –651.

[Zhu, 2003] J. Zhu, A. Hoorfar, and N. Engheta, “Bandwidth, cross-polarization, and feed-point characteristics of matched Hilbert antennas“, IEEE Antennas and Wireless Propagat. Let. , vol. 2, no. 1, 2003, pp. 2–5.

- 15 -

DISCLAIMER

The work associated with this report has been carried out in accordance with the highest technical standards and the FRACTALCOMS partners have endeavoured to achieve the degree of accuracy and reliability appropriate to the work in question. However since the partners have no control over the use to which the information contained within the report is to be put by any other party, any other such party shall be deemed to have satisfied itself as to the suitability and reliability of the information in relation to any particular use, purpose or application. Under no circumstances will any of the partners, their servants, employees or agents accept any liability whatsoever arising out of any error or inaccuracy contained in this report (or any further consolidation, summary, publication or dissemination of the information contained within this report) and/or the connected work and disclaim all liability for any loss, damage, expenses, claims or infringement of third party rights.