## 7 Design of the Directional Antenna  
The geometry we are willing to test is shown in Figure 2. The antenna device is constituted by two conductive cones separated by a little gap. These are supplied by an alternating current via a coaxial cable (not shown in figure), carrying a frequency whose wave-length, in most applications, is of the order of the magnitude of the conductive support size. Each cone features an angle of \(\pi/2\) at the vertex. The need for a gap, possibly very small, between the cones is dictated by the size of the wire connecting to the electrical source. We presume that a gap of approximately 5% of the entire size should not produce significant negative effects, but, of course, this also depends on the degree of directivity we would like to achieve. We consider here the case where the entire antenna is immersed in a region occupied by a linear homogeneous medium \(\Omega\), having phase velocity of the electromagnetic radiation lower than the speed of light in vacuum. Such a medium is characterized by a permittivity coefficient \(\varepsilon_{r}\varepsilon_{0}>\varepsilon_{0}\) and features non-magnetic properties, so that the magnetic permeability coefficient will remain \(\mu_{0}\). In Figure 2, the shape is designed depending on some refraction properties that we are going to describe here below. Our aim is to have a signal emerging from the antenna perfectly horizontally. In this way, the electromagnetic field follows the behavior of Figure 1 (central picture), while it remains in \(\Omega\). After the transition to the outside medium (vacuum or air, in this case), the message follows instead the behavior of the right-hand side picture of Figure 1. In terms of the velocity vector \(\mathbf{V}\), we have the following situation. Inside \(\Omega\), the magnitude of \(\mathbf{V}\) is: \(|\mathbf{V}|=1/\sqrt{\varepsilon_{r}\varepsilon_{0}\mu_{0}}=c/\sqrt{\varepsilon_{ r}}\). In the open space,  
Figure 2: A biconic antenna (upper and lower blue cones) is surrounded by a dielectric of a suitable shape (pink ellipsoid-like body). The aim is to get an output signal where the Poynting vectors are aligned horizontally. The black arrows are initially omnidirectional, but they emerge from the dielectric with zero components along the \(z\)-axis. we have as usual: \(|\mathbf{V}|=c\). At the interface between the two media, we enforce the rules of geometrical optics. We denote by \(\partial\Omega\) the border of \(\Omega\), i.e., the interface between the dielectric and the outside vacuum. The determination of the surface \(\partial\Omega\) follows from the solution of a differential problem as explained hereafter. As shown in Figure 3, in the Cartesian framework \((r,z)\), \(\partial\Omega\) corresponds to the graph of a function \(z=s(r)\) defined through the following constraints. First of all, the curve passes through the point \((a,b)\). We have \(a=b\) when the cones have the vertex angle equal to \(\pi/2\) (here we are neglecting the gap between the cones). A light ray comes from the origin with an angle \(\alpha\). As we said, the curve \(s\) must be designed in such a way that the emerging ray is horizontal. To proceed with the determination of \(s\) we need to recall [PERSON]'s law (see [28], p. 38). In Figure 3, \(\theta_{1}\) denotes the difference between \(\alpha\) and the angle determined by the direction normal to \(\partial\Omega\). Similarly, \(\theta_{2}\) denotes the angle between the normal direction and the abscissa. In this particular situation, [PERSON]'s law says that:  
\[\sqrt{\varepsilon_{r}}\ \sin\theta_{1}\ =\ \sin\theta_{2} \tag{30}\]  
We can translate in formulas the above settings by writing:  
\[\tan(\theta_{2})=-\frac{1}{s^{\prime}(r)}\hskip 28.452756 pt\tan(\alpha)=\frac{s (r)}{r}\hskip 28.452756 pt\tan(\theta_{1})=\tan(\theta_{2}-\alpha)=\frac{r\ +\ s(r)\ s^{\prime}(r)}{s(r)\ -\ r\ s^{\prime}(r)} \tag{31}\]  
where the prime denotes the derivative with respect to the variable \(r\).