Appendix A: Fields from a Fractal Structure

• Fields from a Fractal Antennae
• The Far field

The fields from a line element can be solve with the help of the Hertz Vector [Marion and Heald, 1980]. In order to solve Maxwell's equations we define, in empty space, the vector function $\mathbf{Q}$ [ Marion and Heald, 1980] that is related to the current density $\mathbf{J}$ and the charge density $\rho$ as

$$\mathbf{J}=-\frac{\partial \mathbf{Q}}{\partial t}$$

$$\rho =\mathbf{\nabla }\cdot \mathbf{Q}$$

Note that $\mathbf{Q}$ solves the continuity equation trivially, and furthermore, it can be used to define another vector function, namely the Hertz vector $\mathbf{\Pi }(\mathbf{x},t),$ as

$$\nabla ^2\mathbf{\Pi -}\frac 1{c^2}\frac{\partial ^2\mathbf{\Pi }}{\partial t^2}=-4\pi \mathbf{Q}$$

where the fields are then defined as

$$\mathbf{B}(\mathbf{x},t)=\frac 1c\mathbf{\nabla }\times \frac{\partial \mathbf{\Pi }(\mathbf{x},t)}{\partial t}$$

$$\mathbf{E}(\mathbf{x},t)=\mathbf{\nabla }\times \mathbf{\nabla }\times \mathbf{\Pi }(\mathbf{x},t)$$

The time-Fourier transformed Maxwell's equations, with $\mathbf{Q}(\mathbf{x} ,\omega \mathbf{)=}\frac i\omega \mathbf{J}(\mathbf{x},\omega )$ and $\mathbf{J}(\mathbf{x},\omega )=\widehat{\mathbf{L}}\mathrm{I}(l,\omega ),$can be solved with the help of the Hertz vector $\mathbf{\Pi }$,

$$\mathbf{\Pi }(\mathbf{x},\omega )=\frac i\omega \int_0^L\mat... ... }}{\left\Vert \mathbf{x-}l\widehat{\mathbf{L}}\right\Vert }dl$$ (24)

where the line element has orientation $\mathbf{L}$ and length L, and is parametrized by $l\;\epsilon \;[0,L]$. Values with the hat $\widehat{}$indicate unit vectors, variables in bold indicate vectors, $\omega$ is the frequency, $k=\frac \omega c$. The time dependence can be found by inverting the above equation.

Fields from a Fractal Antennae

A current pulse propagates with speed $\beta =\frac{\mathrm{v}}c$ along a fractal structure. At the n^th line element with orientation $\mathbf{L}_n$ and length L_n, which is parametrized by $l\;\epsilon \;[0,L_n]$, the current is given by $I(l,s_n,t)=I_o(t-\frac{s_n+l}{\mathrm{v}})$ where s_n 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 $\{\mathbf{r}_n,\mathbf{L}_n,I(s_n,t)\,\vert\,n=0,...,N\}$ of line elements, such as shown in the example diagram of Fig. A.1, the Hertz vector is given by  

$$\mathbf{\Pi }(\mathbf{x},\omega )=\sum_{\{n\}}\widehat{\math... ... }}{\parallel \mathbf{r}_n-l\widehat{\mathbf{L}}_n\parallel }dl$$ (25)

where $\mathbf{r}_n$ is the vector from the beginning of the n^th line element to the field position $\mathbf{x}$, $\omega$ is the frequency and $k=\frac \omega c$.

  

Figure A.1: A diagram that explains all the variables and coefficients

We must realize that in general Eq. A.2 for $\mathbf{\Pi }$ is very complicated, but we are interested in the far field of the small line elements ($r_n\gg L$). 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

$$\mathbf{B}(\mathbf{x},\omega )=-\sum_{\{n\}}\frac{k^2e^{ikr_... ...{(kr_n)}](\widehat{\mathbf{L}}_n\times \widehat{\mathbf{r}}_n)$$

$$\begin{array} {c} \mathbf{E}(\mathbf{x},\omega )=-\sum_{\{n\... ...f{r}} _n)(1+\frac{3i}{(kr_n)}+\frac{3i^2}{(kr_n)^2}]\end{array}$$

where the geometric factor is given by

$$f(s_n,\omega ,r_n)=\frac i\omega \int_0^{L_n}I_o(s_n,l,\omeg... ...}-( \widehat{\mathbf{L}}_n\cdot \widehat{\mathbf{r}}_n)k)l}dl$$

$$f(s_n,\omega ,r_n)=\frac{\beta I_o(\omega )e^{i\frac \omega ... ...}-(\widehat{\mathbf{L}}_n\cdot \widehat{\mathbf{r}} _n)k)L_n})$$

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

$$\mathbf{B}(\mathbf{x},t)=\sum_{\{n\}}\frac{-(\widehat{\mathb... ...-\tau _1}+\frac c{r_n}I_1(\tau )\mid _{t-\tau _2}^{t-\tau _1}]$$

 

$$\begin{array} {c} \mathbf{E}(\mathbf{x},t)=\sum_{\{n\}}\frac... ...c^2}{r_n^2}I_2(\tau )\mid _{t-\tau _2}^{t-\tau _1})]\end{array}$$ (26)

where

$$I_1(t)=\int_{-\infty }^td\tau I_o(\tau )$$

$$I_2(t)=\int_{-\infty }^td\tau \int_{-\infty }^\tau d\tau ^{^{\prime }}I_o(\tau ^{^{\prime }})$$

can be calculated exactly for the current described above, and where

$$\tau _1=\frac{r_n}c+\frac{s_n}{\mathrm{v}}$$

$$\tau _2=\frac{r_n+(\widehat{\mathbf{L}}_n\cdot \widehat{\mat... ...widehat{\mathbf{L}}_n\cdot \widehat{\mathbf{r}} _n)\frac{L_n}c$$

The value of $\tau _1$ and $\tau _2$ 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 $I(t)=I_oe^{-\alpha t}(1-\cos (2\pi n\alpha t))\theta (t)$ where $\theta (t)$ is the step function, and $n\geq 1$. Note that the total charge discharged by this current is Q=$I_o/\alpha$ where $1/\alpha$ 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 $\tau =\frac s{\mathrm{v}}+\alpha$ where s is the largest path length along the fractal. The fields are invariant as long as $\alpha t$ , $\frac{L\alpha }{\mathrm{v}}$ and $\mathbf{r}\alpha$ 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 $S(W/m^2)=c\varepsilon _oE^2(V/m),$where 1/$c\varepsilon _o$ is the impedance of free space, as a natural description for the amount of power radiated through a cross-sectional area.

The Far field

The far field is approximately given by  

$$\mathbf{E}(\mathbf{x},t)=\sum_{\{n\}}\frac{\beta I_o(\tau )\... ...(1-\beta (\widehat{\mathbf{L}}_n\cdot \widehat{\mathbf{r}}_n))}$$ (27)

In general we are going to use a current pulse defined as $I(t)=I_o(e^{-\alpha t}-e^{-\gamma t})(1+\cos (\omega t))\theta (t)$ with $\omega =2\pi \alpha n_f$ and $\theta (t)$ as the step function. Here n_f represent the number of oscillations during the decay time scale 1/$\alpha$. We chose the decay parameters as $\alpha =10^3$ s^-1 and $\gamma =2\times 10^5$ s^-1, hence $\gamma /\alpha =200$, which correspond to realistic parameters for lightning [Uman, 1987].

As a measure of the amount of energy radiated to a given point in the far field, we can define an array factor as $R(x,y,z)\sim \alpha \int E^2dt.$From Eq. (A.4 ) we can write this array factor as

$$R\simeq \frac{\beta ^2\zeta ^2}{4(4+5\zeta ^2+\zeta ^4)}\sum... ...)r_nr_m}\{f[\left\vert \tau _n^f-\tau _m^f\right\vert ,\zeta ]+$$

 

$$f[\left\vert \tau _n^i-\tau _m^i\right\vert ,\zeta ]-f[\left... ...,\zeta ]-f[\left\vert \tau _n^i-\tau _m^f\right\vert ,\zeta ]\}$$ (28)

$$f[\tau ,\zeta ]=e^{-\tau }[2+2\zeta ^2+(\zeta ^2-2)\cos (\zeta \tau )+3\zeta \sin (\zeta \tau )]$$

where $\tau _n^i=\alpha (\frac{r_n}c+\frac{s_n}{\mathrm{v}})$ corresponds to the parameters from the beginning (i) of the line element, and similarly for the endpoint (f). Also $\zeta =2\pi n_f$ and I_n is the current strength of the n^th element. The array factor can be normalized by maximum in the array factor corresponding to the single dipole, i.e.,

$$R_o\simeq \frac{\beta ^2\mathrm{I}_o^2\;A}{4(1-\beta a)^2h^2}$$

where $A=\{\frac{3\zeta ^4-\zeta ^2f[\frac 1{\alpha \mathrm{v}}(\beta \Delta r-L),\zeta ]}{2(4+5\zeta ^2+\zeta ^4)}\}\simeq 1,$ with $\Delta r\simeq L\widehat{\mathbf{x}}\cdot \widehat{\mathbf{r}}$ as the difference in distance between the beginning and end points of the dipole to the detector position. h is the height of the detector.