For the purposes of this work, we assume that a fractal antenna can be formed as an array of ''small'' line elements having a fractal distribution in space. Such description is consistent with our understanding of fractal discharges and lightning observations as discussed by LeVine and Meneghini [1978], Niemeyer et al. [1984], Sander [1986], Williams [1988], and Lyons [1994]. Appendix A develops the theory for the calculation of the fields produced by a fractal antenna composed of small line elements and for the calculation of the array factor in the far field of the fractal.
Fractals are characterized by their dimension. It is the key structural
parameter describing the fractal and is defined by partitioning the volume
where the fractal lies into boxes of side 
. We hope that over
a few decades in 
, the number of boxes that contain at least
one of the discharge elements will scale as 
. It is easy to verify that a point will have D=0, a line will have 
and a compact surface will have D=2 . The box counting dimension [
Ott, 1993] is then defined by
For a real discharge there is only a finite range over which the above
scaling law will apply. If 
is too small, then the elements of
the discharge will look like one-dimensional line elements. Similarly, if 
is too large, then the discharge will appear as a single
point. It is, therefore, important to compute D only in the scaling range,
which is hopefully over a few decades in 
. The fractal
dimension will be an important parametrization for the fractal discharge
models that we will explore later, and will impact significantly the
intensity and spatial structure of the radiated pattern.
We consider a fractal antenna as a non uniform distribution of radiating elements (Fig. 1). Each of the elements contributes to the total radiated power density at a given point with a vectorial amplitude and phase, i.e.
The vector amplitudes A
represent the strength and orientation
of each of the individual elements, while the phases 
are in
general related to the spatial distribution of the individual elements over
the fractal, e.g. for an oscillating current of the form e
the
phases vary as 
where 
and r is the position
of the element in the fractal.
Figure 1: A spatially nonuniform distribution of radiators, each
contributiong to the total radiation field with a given phase.
In the sense of statistical optics, we can consider the ensemble average of
Eq. (2), using an ergodic principle, over the spatial
distribution 
of the fractal elements [Goodman, 1985]. For
simplicity we assume that the distributions for each of the elements are
independent, and also the same, hence
By requiring that 
we obtain that
the ensemble average is
If the distribution of the phases is uniform (e.g. random) then 
and 
. On the other hand, if there is perfect coherence we have 
and G=1. In general, a fractal antenna will display a
power law distribution in the phases 
(multiplied by the factor 
so it is finite at the
origin), where 
corresponds to the uniform distribution case and 
corresponds to perfect coherence. Figure 2 shows the plot of 
as a function of 
It can be seen that a power law
distribution of phases, or similarly a power law in the spatial structure,
gives rise to partial coherence.
Figure 2: A plot of 
as a function of 
.
If the distribution of the vector amplitudes does not satisfy the above
relations, e.g. the radiators are oriented in arbitrary directions, then the
power density will be less coherent due to 
. A similar result can be achieved by having a power law
distribution in the amplitudes. In conclusion, the radiation field from a
power law distribution of phases will have a point where the phases from the
radiators will add up almost (partially) coherently showing a significant
gain over a random distribution of phases. Hence the concept of a fractal
antenna.
The partial coherence of the radiators depends on the spatial power law
distribution. Such a power law distribution of phases can be visualized with
the help of Cantor sets [Ott, 1993]. A family of Cantor sets is
constructed by successively removing the middle 
fraction from an
interval, taken as [0,1], and repeating the procedure to the remaining
intervals (see Fig. 3). At the n
step, a radiator is
placed at the mid-point of each of the remaining intervals.
Figure 3: The construction of the fractal distribution of the radiators from
the 
Cantor set.
Note that for 
we obtain a uniform distribution of elements, but
for 
the radiators are non-uniformly distributed, and in fact
the spatial distribution follows a power law that can be described by its
fractal dimension. Suppose that for 
we require 
intervals to cover the fractal, then it is clear that with 
we would require 
intervals to cover the fractal. But the
fractal is the same, therefore, 
. From the scaling 
we obtain
that the dimension is given by
We can go further, and write a formula for the radiation field due to the 
Cantor set of radiators. Note that if at the n
step we have
the radiators placed at the sequence of points 
then at the n
+1 step each radiator at x
will be replaced by two
radiators at 
generating
the sequence 
. Since we start with 
the sequences 
at the n
step are trivially constructed.
The radiation field (see Eq. (1)) from this 
Cantor set
at the n
step can then be written as
where 
, L is the spatial extent of the fractal, 
is the angular position of
the detector, and 
(taken as zero) is the phase of the m
element. The radiators are given a strength proportional to the measure 
(or length) of the segment which defines it.
The space dependence of the radiation fields is plotted in Fig. (4)a-b for 
(
) and 
(D=1)
respectively, where the sets have been taken to the 5
level. The most
relevant issue for our purposes is the fact that there is a direction at
which phases add coherently (partially) for 
while this does not
happen for the homogeneous case 
.
Figure 4: The spatial dependence of the radiation fields for (a) 
,
and (b) 
, D=1.
Therefore, partial coherence occurs naturally in systems that have power-law spatial distributions. We are now ready to turn to the properties of fractal antennae with propagating currents. Specifically, how tortuosity and branching can increase the radiated field intensity in some locations as compared with single dipole antennae.
