## 2 Transforming Wave Polarization Using Arrays of Passive Radiators  
Generating a Dual Polarization Using a Linear Radiator Tilted with Respect to the Direction of the Incident Electric Field  
Consider a TEM wave propagating along the Oz axis (Figure 1a). The electric field intensity vector can be decomposed along the axes O\(x\) and O\(y\), as shown in Figure 1b, and can be written as:  
\[\mathbf{E}=E_{x}(z,t)\hat{a}_{x}+E_{y}(z,t)\hat{a}_{y} \tag{1}\]where  
\[E_{x}(z,t)=E_{x,0}\cos(\omega t-k_{0}z) \tag{2}\]  
\[E_{y}(z,t)=E_{y,0}\cos(\omega t-k_{0}z+\Delta\Phi) \tag{3}\]  
\[k_{0}=\frac{2\pi}{\lambda} \tag{4}\]  
and \(\Delta\Phi\) is the initial phase shift between the two components of the electric field.  
If \(\Delta\Phi=\frac{\pi}{2}\) then:  
\[\mathbf{E}=E_{x,0}\hat{a}_{x}\cos(\omega t-k_{0}z)+E_{y,0}\hat{a}_{y}\sin( \omega t-k_{0}z) \tag{5}\]  
and if, in addition, \(E_{x,0}=E_{y,0}=E_{0}\) then  
\[\mathbf{E}=E_{0}\big{[}\hat{a}_{x}\cos(\omega t-k_{0}z)+\hat{a}_{y}\sin(\omega t -k_{0}z)\ \big{]} \tag{6}\]  
In that case, \(\mathbf{E}=\mathrm{ct}\), and the polarization of the wave is circular.  
Now let us consider a linearly polarized wave along the Oz axis, incident on an elementary dipole located in the plane (\(x\)O\(y\)), inclined at \(45^{\circ}\) with respect to the O\(y\) axis (Figure 2). The radiator is considered to have a circular section of finite radius, \(a\ll\lambda\) and an infinitesimal length ds'.  
By applying the boundary conditions, the superficial current density induced on the surface of the radiator will be:  
\[\mathbf{J_{s}}=\frac{2}{\sqrt{2}}\hat{n}\times\mathbf{H}_{i} \tag{7}\]  
and the induced current can be found by integrating the current density on the side surface of the radiator:  
\[I=2\pi a\frac{2H_{i}}{\sqrt{2}}=4\pi a\frac{E_{i}}{\eta\sqrt{2}} \tag{8}\]  
Figure 1: TEM wave propagating along the Oz axis (**a**) and electric field strength components (**b**).  
Figure 2: Linearly polarized wave incident to an elementary dipole.  
where \(\eta\) is the free space wave impedance.  
The reradiated electric field in a direction perpendicular to the current element will be:  
\[E_{r}=-jk_{0}\frac{E_{i}}{\sqrt{2}}\frac{a}{r}\exp(-jk_{0}r)\ \mathrm{d}s^{\prime} \tag{9}\]  
The vector \(E_{r}\) decomposes into two orthogonal components, one along the Ox axis and the other one along the O\(y\) axis, of magnitude  
\[E_{rx}=E_{ry}=-jk_{0}E_{i}\frac{a}{2r}\exp(-jk_{0}r)\mathrm{d}s^{\prime} \tag{10}\]  
The components of the total electric field (incident and reradiated) are shown in Figure 3. Their magnitudes are  
\[E_{x,tot}=-E_{rx}=jk_{0}E_{i}\frac{a}{2r}\exp(-jk_{0}r)\mathrm{d}s^{\prime} \tag{11}\]  
\[E_{y,tot}=E_{i}+E_{ry}=\frac{E_{i}}{2r}\big{(}2r_{0}-jk_{0}\mathrm{d}s^{\prime} a\big{)}\exp(-jk_{0}r) \tag{12}\]  
where \(r_{0}\) is the distance between the primary radiation source, i.e., the horn aperture, and the elementary dipole.  
When \(r_{0}\gg a\), the first term in the expression of \(E_{y,tot}\) is dominant and \(E_{x,tot}\) and \(E_{y,tot}\) will be out of phase with each other by approximately \(\pi/2\).  
The magnitude of the \(E_{rx}\) component can be increased by successively exciting arrays of passive radiators, as is the case for Yagi-Uda antenna arrays. Conversely, such an array of passive radiators would not increase the magnitude of \(E_{ry}\) to the same extent as for \(E_{rx}\), given the magnitude and phase relationship between the terms in \(E_{y,tot}\).