A current pulse propagates with speed 
along a
fractal structure. At the n
line element with orientation 
and length 
, which is parametrized by 
, the
current is given by 
where 
is the path length along the fractal (or if you prefer a phase shift). The
radiation field is the superposition, with the respective phases, of the
small line current elements that form the fractal. For a set 
of line elements, such as shown in
the example diagram of Fig. 25, the Hertz vector is given by
where 
is the vector from the beginning of the 
line
element to the field position 
, 
is the frequency and 
.
Figure 25: A diagram that explains all the variables and coefficients
We must realize that in general Eq. 10 for 
is
very complicated, but we are interested in the far field of the small line
elements (
). Therefore, we can take the far field approximation of
the small line elements to obtain a closed form solution for the Fourier
transformed fields as
where the geometric factor is given by
Note that even though we are in the far field of the small line elements, we can be in fact in the intermediate field with respect to the global fractal structure. Therefore, phase correlations over the fractal can be extremely relevant, and produce spatially nonuniform radiation fields. We then invert the Fourier transform of the field to real time and obtain the spatio-temporal radiation pattern due to the fractal discharge structure
where
can be calculated exactly for the current described above, and where
The value of 
and 
correspond to the causal time delays
from the two end points of the line element.
Before finishing this section we want to mention that there is an inherent
symmetry in the radiation fields. In general we will assume that the current
is given by 
where
is the step function, and 
. Note that the total charge
discharged by this current is Q=
where 
is the decay
time of the current. But since the current propagates along the fractal, the
radiation fields at a given position in space will last for a time given by 
where s is the largest path length along
the fractal. The fields are invariant as long as 
, 
and 
are kept constant in the
transformation. Such scaling can become relevant in studying the properties
of radiation fields from fractal antennae.
In general we will use the power density 
where 1/
is the impedance of free space, as a natural
description for the amount of power radiated through a cross-sectional area.
