To illustrate the properties of fractal antennae compared to those of simple
dipole radiators, we take the fractal antenna as composed of small line
elements and compute its far field radiation pattern. For an oscillating
current 
that propagates with speed 
v/c
along the antenna, the contribution from each line element to the total
radiation field is (from Eq. (1))
where 
, 
is the position of the beginning of the line element from the origin, and 
. Radiation occurs when
there is a change in the direction of the propagating current. Also note
that mathematically we can describe a radiator with a nonpropagating current
in the non-physical limit 
.
In general, the radiation pattern of an antenna can be effectively excited,
only by certain frequencies corresponding to the characteristic length
scales of the antenna, e.g. 
(see Eq. (4)). Therefore,
if there is no characteristic size, as in the case of a power law structure,
then the antenna will generate an effective radiation pattern for a whole
range of frequencies controlled by the smaller and largest spatial scale.
Such antenna is called a broad band antenna, and that is why fractal
antennae are so important in many applications.
By spatially superposing these line radiators we can study the properties of simple fractal antennae. Of special interest, to our high altitude lightning work, is to compare the radiation pattern of these fractal models with a simple (meaning one line element) dipole antenna.
