# A Phased Array Antenna with New Elements Designed Using Source Transformations
## 2 Theoretical Model of \"Pinwheel\" Antenna Array  
Next, consider the N-element dipole phased array along the y-axis represented in Figure 1b (left) and positioned in free-space, denoted as the \"reference array\". Each of the elements in the \"reference array\" are equally spaced with the edge-to-edge distance between the elements of \(g=\lambda/15\), where \(\lambda\) is the free-space wavelength at which the phased array is designed to operate. A two dimensional (2D) space is considered to illustrate the proposed \"pinwheel\" array. The current distribution of each of the elements from the \"reference array\" is defined by \(\mathbf{I}_{n}=J.e^{i(n-1)\varnothing}\), where \(\mathbf{J}\) is used to approximate the current distribution on a thin wire at \(x=0\) and \(n=1,2\ldots\mathrm{N}\), and \(\varnothing\) is the phase between the adjacent antenna elements. In this case, an array of four elements is chosen to validate the proposed method. Figure 1: Proposed material-embedded antenna array using TO technique: (**a**) “pinwheel” transformation of a single dipole antenna element with TO-embedded media; (**b**) transformation of linear dipole array (reference array) (left) into a linear array of “pinwheel” antenna elements (right). Each of the dipoles in the \"reference array\" is of \(\lambda/2\) length spanning over a distance of \(2.2\lambda\). We intend to transform the reference array into a linear array of equidistant complex geometry antennas, where each antenna element of the \"reference array\" will be transformed into a \"pinwheel\"-shaped antenna, as shown in Figure 1b. The first task is to transform the individual dipole elements of the \"reference array\" into a \"pinwheel\"-shaped antenna element. The current distribution of a dipole in Figure 1a along \(x=0\) was chosen to be [14]:  
\[\overline{J}=\left[ \begin{array}{cc}0&\\ \frac{1}{\sqrt{\sigma*\pi}}.\ e^{-\frac{2}{\sigma}}\\ 0&\\ \end{array} \right] \tag{1}\]  
where the limit \(\sigma\to 0\) can be taken to arrive at a sheet current density [8] and is set to be infinitesimally small relative to the length of the dipole. The half-wave dipole is initially defined by its current distribution in Cartesian coordinates and is converted into cylindrical coordinates to facilitate the transformation. The material parameters for the space under transformation is given by [19]:  
\[\epsilon^{\prime}=\mathbf{\mu}^{\prime}=\ \frac{A\epsilon A^{T}}{detA} \tag{2}\]  
where \(A=\ \partial(x^{\prime},y^{\prime},z^{\prime})/\partial(x,\ y,\ z)\) is the Jacobian matrix and \(A^{T}\) is transpose of the Jacobian matrix. The transformed current density, \(\overline{J}^{\prime}\), under coordinate transformation is given by [12]:  
\[\mathbf{J}^{\prime}=A|J|/|A| \tag{3}\]  
The current density was transformed into \"pinwheel\" coordinates from cylindrical coordinates using (3) and by adopting the \"pinwheel\" coordinate transformation introduced in [14] as:  
\[\rho^{\prime}=\rho \tag{4}\]  
\[\theta^{\prime}=\left\{ \begin{array}{ccc}\theta&,&\rho>R_{1}\\ \theta+\Delta\theta\ \Big{(}1-\frac{\rho}{R_{1}}\Big{)}&,&\rho<R_{1}.\\ \end{array} \right. \tag{5}\]  
In the \"pinwheel\" transformation, the coordinates were rotated by an angle, \(\Delta\theta\), between the origin and \(\rho=\ R_{1}\). Current sources in the region \(\rho<R_{1}\) were chosen (i.e., \(R_{1}\) is radius of the material shell). To retrieve the current distribution in the Cartesian (original) coordinate system, we perform the inverse of the cylindrical transformation. By using the inverse relations of cylindrical and Cartesian coordinates, Equation (3) yields the final transformed current distribution as:  
\[\overline{\mathbf{J}^{\prime}}=\frac{1}{R_{1}\sqrt{\sigma*\pi}}exp\left(-\frac{( \rho\cos(v))^{2}}{\sigma}\right)*\left( \begin{array}{c}-R_{1}\big{(}\sin(v+ \theta)+\rho\Delta\theta sin\theta\sin\Big{(}\Delta\theta\Big{(}\frac{\rho}{ R_{1}}-1\Big{)}+\theta\big{)}\\ R_{1}\cos(v+\theta)+\rho\Delta\theta cos\theta\sin(v)\\ 0\\ \end{array}