WHEN Nathan 鈥淐hip鈥� Cohen decided to set up a ham radio system at home, he hit a snag. The lease for his flat in the centre of Boston stipulated that he could not erect an antenna outside the building. Without an antenna he couldn鈥檛 send or receive radio signals. A small problem, but the answer he came up with has changed his life.
Instead of using a conventionally shaped antenna, Cohen made something entirely different. He cut a sheet of aluminium foil into the shape of a mathematical pattern known as an inverse Koch curve and stuck the pattern onto a sheet of A4-sized paper. An inverse Koch curve is a fractal that looks like a series of triangles stacked up on top of each other like a pagoda. Like all fractals it is 鈥渟elf similar鈥�-it appears the same regardless of the scale at which it is viewed.
Cohen connected the foil to his radio receiver to see if it might serve as a covert antenna if he mounted it outdoors. To his surprise, the fractal foil pattern worked well and for a while Cohen was able to continue his hobby without arousing suspicion. 鈥淚t didn鈥檛 seem very revolutionary at the time.
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That was in 1988. Today, Cohen鈥檚 experiment has made him a pioneer in the new field of fractal antenna design. It turns out that fractal antennas have many advantages over their conventional counterparts. For a start, they are smaller-a fractal antenna for a mobile phone can be made the size and shape of a 35-millimetre photographic slide and can be built into the casing. It could even be printed like an integrated circuit. Fractal antennas are also better at picking up signals and can receive them over a wider range of frequencies. But there鈥檚 a challenge ahead. While the new antennas are set to be used in everything from mobile phones to huge receiving arrays, physicists are being left behind. Nobody is exactly sure why fractals make such good antennas. Now the race is on to find out.
Strike a chord
Antennas work rather like the strings on a piano. When struck, a piano string vibrates at a specific wavelength. Because the ends of the string are fixed and cannot move, the wavelength must be some multiple of this distance. The simplest resonance will have a wavelength that is twice the length of the string. A similar effect occurs when a conducting wire is 鈥渟truck鈥� by radio waves. The waves induce a variable current along the length of the wire, and since this current must be zero at the ends, the wavelength of any current fluctuations can only be some multiple of the wire鈥檚 length. And the longer the wavelength, the longer the antenna must be to receive it.
In practice, the range of frequencies an antenna can broadcast and receive can be varied by changing the electrical properties of the circuits to which it is connected. Looping the antenna or adding small perpendicular wires to it also changes properties such as its capacitance and inductance. It is even possible to predict the performance of certain shapes using equations that describe the electromagnetic behaviour of materials.
These equations were developed in the 1800s by James Clerk Maxwell. 鈥淵ou can solve Maxwell鈥檚 equations fairly straightforwardly for uniform curvilinear antennas-that is, things like loops-or straight wires,鈥� says Cohen, who is now chief technical officer of Fractal Antenna Systems in Fort Lauderdale, Florida, and a professor at Boston University. The challenge is to come up with a way of solving Maxwell鈥檚 equations for fractal patterns. Nobody has yet succeeded.
Making fractals is not difficult. A fractal 鈥済rows鈥� through a series of steps, or iterations. A Minkowski box fractal, for example, begins as a straight line. The first iteration adds a box with its base removed to the middle of the line to create a shape like a square wave. The second iteration repeats this process in the middle of every straight line in the shape. This adds a further five smaller boxes. The third iteration repeats this process again, and so on ad infinitum (see Diagram). Making a fractal antenna requires a single wire that is bent many times to make the required shape. This bending makes the antenna much more compact.
Despite the tradition of using simple shapes for antennas, Cohen returned to the idea of fractal antennas in the 1990s. Starting with a straight-wire antenna, he tested its 鈥済ain鈥�, which is a measure of how well it transmits a signal in a beam. Then he bent it into the first iteration of a Minkowski box fractal-the square-wave shape. The gain increased by four decibels. Since the decibel scale is logarithmic, says Cohen, that鈥檚 a substantial amount of gain. 鈥淏ut that wasn鈥檛 too much of a surprise. People had added stubs to antennas like that before.鈥�
The big surprise came when Cohen added the second and third iterations. To his amazement, the gain remained the same as that for the square-wave shape, even though the antenna became more compact.
A fractal antenna鈥檚 resonances-the wavelengths to which it responds-also change as iterations are added. And in a way that is hard to explain. Researchers believe that a number of processes are at work. The iterations add smaller line segments to the fractal, and each of these can act like an individual antenna that responds to shorter wavelengths. In addition, the iterations add bends to the antenna and this changes its capacitance and inductance. This process is called fractal loading and the result is that the antenna resonates at both shorter and longer wavelength signals. Because of this, the range over which the antenna can receive signals-its bandwidth-grows.
While increasing the number of iterations makes the antennas smaller without reducing gain, there is a practical limit to how small they can get. This is because the diameter of the wire must also get smaller to accommodate the tiny bends. Smaller wires have higher electrical resistances, making them less efficient at picking up and sending radio signals. 鈥淭here are certainly diminishing returns-on most antennas-for iterations above, I鈥檇 say, five or six,鈥� says Cohen.
Maxwell鈥檚 dilemma
Higher-order iterations create problems for physicists too. Calculating an antenna鈥檚 performance with Maxwell鈥檚 equations depends on the distribution of electrical current along it. For simple shapes, like lines and loops, the equations can be easily solved. But no solutions exist for most fractals, and scientists must use what are known as numerical approaches to find the current distribution. Numerical methods are far from perfect because they make assumptions about the way the antenna works. One way of doing this is to assume that each segment operates as an independent straight-line antenna. 鈥淵ou divide the antenna up, and you find the current on each length,鈥� explains Dwight Jaggard, an electrical engineer at the University of Pennsylvania.
This approach works well for a few iterations, but higher-order iterations contain large numbers of segments that vary in size over many orders of magnitude. The numerical techniques simply cannot cope. Just how this can be solved is not clear, but Doug Werner, a mathematician at Pennsylvania State University, has some ideas. 鈥淵ou might be able to take advantage of the scaling in some clever way to avoid doing the numerical computations at every scale,鈥� he says. It is possible that the first few iterations could reveal a pattern that can be applied to additional iterations. But so far no such methods exist and the performance of many fractal antennas can only be assessed after they have been built.
Large cigars
While some researchers continue to improve their numerical methods, others are attempting to incorporate fractal antennas in real devices. In the near future, says Cohen, fractal antennas will be used inside cellular and cordless phones, replacing the conventional wands. Fractal Antenna Systems has developed the 鈥渇ractal micropatch鈥�, which is smaller than a 35-millimetre slide and about as thick. It can simply be stuck inside the casing of a phone. Previously, the smallest antenna that could work inside a cellular phone was the size of a large cigar.
In a year or two, says Cohen, 鈥測ou will see fractal antennas as part of wireless devices in things like electricity meters and vending machines.鈥� These devices will communicate through cellular phone lines to report meter readings or the need for restocking. Small antennas could even reduce the bulk of equipment carried by military radio operators in the field. Soldiers of the future might even keep in touch with their operational base through fractal antennas built into their helmets.
These little devices, dreamt up to beat an antenna ban, look set to revolutionise the design of radio transmitters and receivers. Perhaps one day they could do the same for the leases on flats in Boston.
